VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES

MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON

1. Introduction

Let ρ̄: G_{Q} → GL_{2}(k) be an absolutely irreducible modular Galois representation over a finite field k of characteristic p. Assume further that ρ̄ is p-ordinary and p-distinguished in the sense that the restriction of ρ̄ to a decomposition group at p is reducible and non-scalar. The Hida family H(ρ̄) of ρ̄ is the set of all p-ordinary p-stabilized newforms f with mod p Galois representation isomorphic to ρ̄. (If ρ̄ is unramified at p, then one must also fix an unramified line in ρ̄ and require that the ordinary line of f reduces to this fixed line.) These newforms are a dense set of points in a certain p-adic analytic space of overconvergent eigenforms, consisting of an intersecting system of branches (i.e. irreducible components) T(𝔞) indexed by the minimal primes 𝔞 of a certain Hecke algebra.

To each modular form f ∈ H(ρ̄) one may associate the Iwasawa invariants μ^{an}(f), λ^{an}(f), μ^{alg}(f) and λ^{alg}(f). The analytic (resp. algebraic) λ-invariants are the number of zeroes of the p-adic L-function (resp. of the characteristic power series of the dual of the Selmer group) of f, while the μ-invariants are the exponents of the powers of p dividing the same objects. In this paper we prove the following results on the behavior of these Iwasawa invariants as f varies over H(ρ̄).

Theorem 1. Fix ∗ ∈ {alg, an}. If μ^{∗}(f_{0}) = 0 for some f_{0} ∈ H(ρ̄), then μ^{∗}(f) = 0 for all f ∈ H(ρ̄). (We then write simply μ^{∗}(ρ̄) = 0.)

It is conjectured by Greenberg that if a p-ordinary modular form f of weight two has a residually irreducible Galois representation, then μ^{an}(f) = μ^{alg}(f) = 0; Theorem 1 thus shows that this conjecture is equivalent to the corresponding conjecture for modular forms of arbitrary weight.

Theorem 2. Fix ∗ ∈ {alg, an} and assume that μ^{∗}(ρ̄) = 0. Let f_{1}, f_{2} ∈ H(ρ̄) lie on the branches T(𝔞_{1}), T(𝔞_{2}) respectively. Then

(1.1)  λ^{∗}(f_{1}) − λ^{∗}(f_{2}) = ∑_{ℓ|N_{1}N_{2}} e_{ℓ}(𝔞_{2}) − e_{ℓ}(𝔞_{1});

here the sum is over all primes dividing the tame level of f_{1} or f_{2} and e_{ℓ}(𝔞_{j}) is a certain explicit non-negative invariant of the branch T(𝔞_{j}) and the prime ℓ.

The first (resp. second) theorem corresponds to Theorems 3.7.5 and 4.3.3 (resp. Theorems 3.7.7 and 4.3.4) with trivial twist by the mod p cyclotomic character.

Note that the right-hand side of (1.1) is identical both algebraically and analytically. In particular, using work of Kato on the main conjecture of Iwasawa theory for modular forms, we obtain the following result.

Corollary 1. Assume that μ^{alg}(ρ̄) = μ^{an}(ρ̄) = 0. If the main conjecture holds for some f_{0} ∈ H(ρ̄), then it holds for all f ∈ H(ρ̄).

The corollary in particular reduces the main conjecture to the case of modular forms of weight two, together with the conjecture on the vanishing of the μ-invariant. We also obtain the following result on the variation of λ-invariants in a Hida family.

Corollary 2. Fix ∗ ∈ {alg, an} and assume that μ^{∗}(ρ̄) = 0.

(1) λ^{∗}(·) is constant on branches of H(ρ̄).

(2) λ^{∗}(·) is minimized on the branches of H(ρ̄) of minimal tame level.

Although we give an extensive summary of our methods in the following sections, we give a brief indication now of our approach. On the analytic side, our results are obtained by specialization of a two-variable p-adic L-function. Our construction is inspired by those of [22, 28], although we are able to obtain somewhat better control over the canonical periods. This control of canonical periods in families and over varying tame level (see Section 3 especially Theorem 3.6.2) is the key technical input needed for our Iwasawa theoretic applications. On the algebraic side we use a theory of residual Selmer groups to relate the Iwasawa invariants of the modular forms in H(ρ̄).

This work was motivated by the paper [15] of Greenberg and Vatsal where Theorems 1 and 2 were obtained for p-ordinary modular forms of weight two. Our results here help to illuminate the results of [15]; indeed, if two congruent elliptic curves have different λ-invariants, it follows from these results that they must lie on two branches of the associated Hida family with different ramification behavior. One may thus think of the change of λ-invariants in terms of “jumps” as one moves from one branch to another at crossing points (which necessarily occur at non-classical^{1} eigenforms).

1.1. Iwasawa theory. The papers [12] of Greenberg and [27] of Mazur introduce the following point of view on Iwasawa theory: If X is a p-adic analytic family of p-adic representations of G_{Q} interpolating a collection of motivic representations, then there should exist an analytic function (or perhaps a section of an invertible sheaf) L, which we call a p-adic L-function, defined on X and a coherent sheaf H on X, such that the zero locus of L coincides with the codimension one part of the support of H (thought of as a Cartier divisor on X). The function L should p-adically interpolate the (suitably modified) values at s = 1 of the classical Lfunctions attached to the Galois representations at motivic points of X, while at such a motivic point x ∈ X, the fiber H_{x} should coincide with an appropriately defined Selmer group of the motive attached to x.

The L-function considered above is what is usually known as the analytic p-adic L-function in classical Iwasawa theory; the equation of the codimension one part of the support of H is usually known as the algebraic p-adic L-function. The equality of the zero locus of L with the divisorial part of the support of H is thus a statement of the main conjecture of Iwasawa theory for the family of Galois representations X.

In this paper, we restrict to the case of two dimensional nearly ordinary modular representations, in which case the space X can be described via Hida theory. Consider a nearly ordinary residual representation; such a representation may be written in the form ρ̄ ⊗ ω^{i} : G_{Q} → GL_{2}(k), where ρ̄ is as above, ω denotes the mod p cyclotomic character and 0 ≤ i ≤ p − 1. As ρ̄ is ordinary, we may (and do) fix a p-stabilization of ρ̄ ⊗ ω^{i}: a choice of one dimensional G_{p}-invariant quotient of ρ̄ ⊗ ω^{i}.

1To avoid circumlocutions, we adopt the convention that weight one is a non-classical weight.

Consider the universal deformation space Y_{Σ}^{n.ord} parameterizing all nearly ordinary deformations of the p-stabilized representation ρ̄⊗ ω^{i} which are unramified at primes in Σ ∪ {p} for some fixed finite set of primes Σ. Any such deformation is of the form ρ ⊗ ω^{i} ⊗ χ, where ρ is an ordinary deformation of ρ and χ is a character of Γ := 1 + pZ_{p}. (We regard χ as a Galois character by composing it with the projection onto Γ of the p-adic cyclotomic character.) If we let Y_{Σ}^{ord} denote the universal deformation space parameterizing all ordinary deformations of the p-stabilized representation ρ̄ which are unramified at primes in Σ∪{p}, then Y_{Σ}^{n.ord} is equal to the product (as formal schemes) of Y_{Σ}^{ord} and Spf (Z_{p}[[Γ]]) . By results of Wiles and Taylor-Wiles [37, 34], as strengthened by Diamond [6], the deformation spaces Y_{Σ}^{n.ord} can be identified with various local pieces of the universal ordinary Hecke algebra constructed by Hida (at least under mild hypotheses on ρ̄).

If we allow Σ to vary, the spaces Y_{Σ}^{n.ord} form a formal Ind-scheme over W (k) parameterizing nearly ordinary deformations of ρ̄ of arbitrary conductor. Let Y^{n.ord} be an irreducible component of this formal Ind-scheme; we may write Y^{n.ord} as the product of Y^{ord} (an irreducible component of the the formal Ind-scheme parameterizing ordinary deformations of ρ̄) and Spf (Z_{p}[[Γ]]). The motivic points are dense on Y^{n.ord} and all have the same tame conductor, which we will denote by N. If T_{N}^{new} denotes the new-quotient of the Hida Hecke algebra of level N, then Y^{ord} corresponds to a minimal prime 𝔞 of T_{N}^{new}; we set X^{ord} = Spf (T_{N}^{new}/𝔞) and X^{n.ord} = Spf (T_{N}^{new}/𝔞 [[Γ]]). (We point out that X^{ord} is nearly the same as Y^{ord} – and hence X^{n.ord} is nearly the same as Y^{n.ord}. Indeed, X^{ord} is a finite cover of Y^{ord} and maps isomorphically to it after inverting p.)

The generalities of Iwasawa theory discussed above apply to this space X^{n.ord}. In Section 3 we give a construction of the analytic p-adic L-function on X^{n.ord}; as is usual, we regard it as a “function of two-variables” – one variable moving along X^{ord} and the other along Spf (Z_{p}[[Γ]]) . An important feature of the situation we consider is that the nearly ordinary deformation space contains many different irreducible components. The main goal of this paper is to describe how to pass Iwasawa theoretic information (such as knowledge of the main conjecture) between components.

In our discussion of Selmer groups we do not go so far as to construct a coherent sheaf of Selmer groups over the space X^{n.ord}. For this, the reader should refer to recent and forthcoming work of Ochiai [30] in which a “two-variable main conjecture” is formulated for each fixed tame level. Moreover, Ochiai shows that this two-variable main conjecture is equivalent to the one-variable main conjecture for each classical modular form in the family. In particular, he deduces our Corollary 1 for modular forms of a fixed tame level. His method has the advantage that he does not need any assumption on the μ-invariant, but has the disadvantage that it only applies to forms on a fixed branch of the Hida family.

1.2. Two-variable p-adic L-functions. To establish the analytic parts of Theorems 1 and 2, we make use of a two-variable p-adic L-function (playing the role of L above). As already remarked upon in 1.1, one variable is a parameter of wild character space (i.e. the standard cyclotomic variable) and the second variable runs through the Hida family (which is a finite cover of weight space) of some fixed residual representation ρ̄. Of the many constructions of two-variable p-adic L-functions (see [14], [33], [31]), our construction follows most closely that of Mazur [28] and Kitigawa [22], with two main differences.

The first difference is that by using the fact that the Hecke algebras are known to be Gorenstein (as in [37]) we may construct these L-function with fewer assumptions on ρ̄. The second difference is that we do not limit ourselves to working solely on a part of the Hida family parameterizing newforms of some fixed tame level.

For each branch Y of the Hida family we construct a two-variable L-function along Y, which at each classical point specializes to the p-adic L-function of the corresponding newform computed with respect to its canonical period. (Actually, the two-variable L-function is defined not on Y, but on the partial normalization X of Y discussed above in 1.1.) However, for any finite set of primes Σ not containing p, we also construct a two-variable L-function along Y_{Σ} (the Hida family of ρ̄ with ramification only at the primes of Σ). If we specialize this two-variable p-adic Lfunction at a classical point, we obtain a non-primitive p-adic L-function attached to the corresponding newform (i.e. the usual p-adic L-function stripped of its Euler factors at primes of Σ). In fact, we prove that this non-primitivity occurs in families: the p-adic L-function on the Hida family Y_{Σ}, restricted to some branch Y, is equal to the p-adic L-function of that branch stripped of its two variable Euler factors at the primes of Σ.

With this construction in hand, the analytic part of Theorem 1 follows immediately. Indeed, the vanishing of the analytic μ-invariant for any particular form f in the Hida family is equivalent to the vanishing of the μ-invariant of the two-variable p-adic L-function along the branch Y containing f, which in turn is equivalent to the vanishing of the μ-invariant of the non-primitive two-variable L-function on Y_{Σ} for any sufficiently large choice of Σ (i.e. such that Y_{Σ} contains Y). This last condition is independent of f and so is equivalent to the vanishing of the μ-invariant for every modular form in the family.

The analytic part of Theorem 2 also follows from this construction. The analytic λ-invariant of a modular form f is equal to the λ-invariant of the two-variable Lfunction attached to the branch containing f. To compare the λ-invariants of two branches, we choose Σ large enough so that Y_{Σ} contains both of these branches. We then relate each λ-invariant to the λ-invariant of the two-variable p-adic L-function attached to Y_{Σ}. As explained above, the difference of the two λ-invariants is then realized in terms of the λ-invariants of certain Euler factors. The quantity e_{ℓ}(𝔞_{i}) appearing in the formula of Theorem 2 is precisely the λ-invariant of the Euler factor at ℓ along the branch corresponding to 𝔞_{i}.

1.3. Residual Selmer groups. Let ρ̄ : G_{Q} → GL_{2}(k) be as above. For any f ∈ H(ρ̄), say with Fourier coefficients in the finite extension K of Q_{p}, Greenberg has defined the Selmer group Sel(Q_{∞}, ρ_{f}) as the subgroup of H^{1}(Q_{∞}, A_{f}) cut out by local conditions, all of which are as strong as possible except for the condition at p; here Q_{∞} is the cyclotomic Z_{p}-extension of Q and A_{f} is (K/O_{K})^{2} with Galois action via ρ_{f}. If one defines Sel(Q_{∞}, ρ̄) in the analogous way, one is confronted with the fundamental problem that the π-torsion on Sel(Q_{∞}, ρ_{f}) may be larger than Sel(Q_{∞}, ρ̄); this is, of course, precisely the reason why congruent modular forms need not have isomorphic Selmer groups.

In [15] this issue is overcome by introducing non-primitive Selmer groups of ρ̄; essentially, if f_{1}, f_{2} ∈ H(ρ̄) have tame levels N_{1} and N_{2} respectively, then the π-torsion on Sel(Q_{∞}, ρ_{f_{1}}) and Sel(Q_{∞}, ρ_{f_{2}}) can both be compared to the nonprimitive Selmer group of ρ̄ obtained by ignoring the local conditions at primes dividing N_{1}N_{2}.

Although this approach can also be made to work in the higher weight case, we proceed somewhat differently. Following Mazur [26], we allow non-strict, but not necessarily vacuous, local conditions S on the cohomology of ρ̄, resulting in a family of residual Selmer groups Sel_{S}(Q_{∞}, ρ̄). We show that for any f ∈ H(ρ̄) there is a local condition S(f) such that

Sel_{S(f)}(Q_{∞}, ρ̄) ≅ Sel(Q_{∞}, ρ_{f})[π].

It follows that the Iwasawa invariants of f can be recovered from the residual Selmer group Sel_{S(f)}(Q_{∞}, ρ̄).

In fact, if the tame level of f coincides with the conductor of ρ̄, then we show that Sel_{S(f)}(Q_{∞}, ρ̄) agrees with the naive residual Selmer group Sel(Q_{∞}, ρ̄). Since such f always exist by level lowering, we are then able to use duality results to show that the difference

dim_{k} Sel_{S(f)}(Q_{∞}, ρ̄) − dim_{k} Sel(Q_{∞}, ρ̄)

is precisely given by the dimensions of the local conditions S(f). The main algebraic results follow from this.

We note that, unlike on the analytic side, Hida theory plays little overt role in these algebraic computations. In particular, in [36], these methods have been applied more generally for algebraic groups other than GL_{2}. The key inputs are automorphic descriptions of the Galois representation at ramified primes, level lowering results and the fact that the Selmer groups of interest are Λ-cotorsion; the remainder of the argument is essentially formal.

1.4. L-functions modulo p. In the paper [25], Mazur raises the question of whether one can define a mod p L-function attached to the residual representation ρ̄ and a choice of tame conductor N (divisible by the tame conductor of ρ̄). The construction discussed in 1.2 gives a positive answer to this question, with the caveat that the appropriate extra data is not simply a choice of tame conductor N, but rather the more precise data of a component Y of the universal deformation space of ρ̄. We may then specialize the p-adic L-function at the closed point of the partial normalization X of Y on which it is defined, so as to obtain a mod p L-function attached to ρ̄ and Y.

We also show that the dimension of Sel_{S(f)}(Q_{∞}, ρ̄) over k depends only on the component Y of the Hida family that contains f; we thus simply write λ^{alg}(Y) to denote this dimension. Assuming that both the analytic and algebraic μ-invariant of ρ̄ vanish, Theorem 2 then shows that the main conjecture for any member of H(ρ̄) is equivalent to the following mod p main conjecture (for one, or equivalently every, choice of Y):

Mod p Main Conjecture. Let Y be a branch of H(ρ̄). Then the mod p L-function L_{p}(Y) of Y is non-zero, λ^{alg}(Y) is finite and

λ(L_{p}(Y)) = λ^{alg}(Y).

1.5. Overview of the paper. In the following section, we recall the Hida theory used in this paper. In particular, we discuss the Hida family attached to a residual representation ρ̄, its decomposition into irreducible components and the various Galois representations attached to this family with an emphasis on the integral behavior.

In the third section, we construct two-variable p-adic L-functions on the Hida family of ρ̄ and on each of its irreducible components. By relating these two constructions to each other and to the construction of classical p-adic L-functions, we obtain proofs of the analytic parts of Theorems 1 and 2.

In the fourth section, we prove the algebraic parts of Theorems 1 and 2 via the theory of residual Selmer groups and the use of level lowering to reduce to the minimal case.

In the final section we give applications to the main conjecture. We discuss some explicit examples to illustrate the general theory, including the congruence modulo 11 between the elliptic curve X_{0}(11) and the modular form Δ.

1.6. Acknowledgments. The authors wish to thank Ralph Greenberg, Masato Kurihara, Barry Mazur and Chris Skinner for helpful conversations.

Notation. We fix an algebraic closure Q̄ of Q and write G_{Q} to denote the Galois group of Q̄ over Q. If Σ is a finite set of places of Q, we write Q_{Σ} for the maximal extension of Q in Q̄ unramified away from Σ. We fix an odd prime p and let Q_{∞} denote the cyclotomic Z_{p}-extension of Q in Q̄. For each prime ℓ of Q (resp. place v of Q_{∞}) fix a decomposition group G_{ℓ} ↪ G_{Q} (resp. G_{v} ↪ G_{Q_{∞}}) with inertia group I_{ℓ} (resp. I_{v}) such that G_{v} ⊆ G_{ℓ} whenever v divides ℓ; note that I_{ℓ} = I_{v} for a place v dividing a prime ℓ ≠ p. Let v_{p} denote the unique place of Q_{∞} above p.

We write ε : G_{Q} → Z_{p}^{×} for the p-adic cyclotomic character. We let Γ denote the group of 1-units in Z_{p}^{×}; the cyclotomic character induces a canonical isomorphism

Gal(Q_{∞}/Q) ⟶^{≅} Γ. If O is a Z_{p}-algebra, we let Λ_{O} denote the completed group ring O[[Γ]]; we simply write Λ for Λ_{Z_{p}}. We denote the natural map Z_{p}^{×} → Z_{p}[[Z_{p}^{×}]] by γ ↦ ⟨γ⟩_{p}; this restricts to the natural map Γ → Λ^{×}.

Recall that Λ is a complete regular local ring of dimension two, non-canonically isomorphic to the power series ring Z_{p}[[T]]. (Such an isomorphism is obtained by choosing a topological generator γ of Γ and mapping ⟨γ⟩_{p} to T + 1.) We let L denote the field of fractions of Λ.

We let Δ denote the group of cyclotomic units of Z_{p}; there is then an isomorphism Γ × Δ ≅ Z_{p}^{×}. We let ω denote the inclusion of Δ in Z_{p}^{×}, so that

Hom(Δ, Z_{p}^{×}) = {ω^{i} | 0 ≤ i ≤ p − 2}.

If Z_{p,(i)} denotes Z_{p} regarded as a Δ-module via the character ω^{i}, then there is a natural isomorphism of Z_{p}[Δ]-algebras Z_{p}[Δ] ≅ ∏_{i=0}^{p−2} Z_{p,(i)}. Combining this with the natural isomorphism Z_{p}[[Z_{p}^{×}]] ≅ Z_{p}[Δ] ⊗_{Z_{p}} Λ, we obtain an isomorphism of Z_{p}[[Z_{p}^{×}]]-algebras

Z_{p}[[Z_{p}^{×}]] ≅ ∏_{i=0}^{p−2} Λ_{(i)},

where Λ_{(i)} denotes a copy of Λ, enhanced to a Z_{p}[[Z_{p}^{×}]]-algebra by having Δ act through ω^{i}.

If N is a natural number prime to p, then we will write Z_{p,N}^{×} := (Z/N)^{×} × Z_{p}^{×}. We will have occasion to consider the completed group ring Z_{p}[[Z_{p,N}^{×}]]. In this situation, we will extend the diamond bracket notation introduced above and write x ↦ ⟨x⟩_{Np} to denote the natural map Z_{p,N}^{×} → Z_{p}[[Z_{p,N}^{×}]]^{×}.

2. Hida theory

2.1. The universal ordinary Hecke algebra. Fix a positive integer N relatively prime to p. We begin by briefly recalling the basic properties of T_{N}, the universal ordinary Hecke algebra of tame level N (denoted by h^{0} (N , Z_{p}) in [17]) acting on ordinary p-adic modular forms of tame level N, arbitrary p-power level and arbitrary weight. We first fix some notation.

For each integer k ≥ 2, let S_{k}(Np^{∞}, R) denote the union over all r ≥ 0 of the spaces of weight k cusp forms on Γ_{1}(Np^{r}) whose Fourier coefficients lie in a fixed Z_{p}-algebra R. We make S_{k}(Np^{∞}, R) into a Z_{p}[[Z_{p}^{×}]]-module by letting Z_{p}^{×} act via the product of the nebentypus action and the character γ ↦ γ^{k}. Let S_{k}(Np^{∞}, R)^{ord} be the subspace of these cusps forms on which U_{p} acts invertibly. If κ is a homomorphisms from Γ = 1 + pZ_{p} to R^{×}, we let S_{k}(Np^{∞}, R)^{ord}[κ] denote the R-submodule of S_{k}(Np^{∞}, R)^{ord} on which Γ acts by the character κ.

If ℘ is a height one prime of Λ, then we set O(℘) := Λ/℘. Let κ_{℘} : Γ → O(℘)^{×} be the character induced by the embedding Γ → Λ^{×}. We say that ℘ is classical of weight k ≥ 2, if it is of residue characteristic zero and if there is a finite index subgroup Γ′ of Γ such that κ_{℘}, when restricted to Γ′, coincides with the character γ ↦ γ^{k} ∈ Z_{p}^{×} ⊂ O(℘)^{×}.

More generally, if ℘ is a height one prime ideal of a finite Λ-algebra T, then we again write O(℘) := T/℘. We say ℘ is classical of weight k, if ℘′ := ℘ ⋂ Λ is classical of weight k and write κ_{℘} := κ_{℘′} : Γ → O(℘′)^{×} ⊂ O(℘)^{×}.

The algebra T_{N} is defined to be commutative algebra of endomorphisms of S_{k}(Np^{∞}, R)^{ord} generated over Z_{p}[[Z_{p}^{×}]] by the Hecke operators T_{ℓ} (resp. U_{ℓ}) for primes ℓ ∤ Np (resp. primes ℓ | Np), together with the diamond operators ⟨a⟩ for a ∈ (Z/N)^{×}. By definition T_{N} is a Z_{p}[[Z_{p}^{×}]]-algebra. The map (Z/N)^{×} → T_{N}^{×} defined via a ↦ ⟨a⟩ allows us to extend this to a structure of Z_{p}[[Z_{p,N}^{×}]]-algebra on T_{N}.

Theorem 2.1.1.

(1) The algebra T_{N} is free of finite rank over Λ = Z_{p}[[Γ]].

(2) If ℘ is a classical height one prime in Λ of weight k, then T_{N}/℘T_{N} is canonically identified with the the quotient of the full Hecke algebra that acts faithfully on S_{k}(Np^{∞}, O(℘))^{ord}[κ_{℘}].

Proof. See [18, Thm. 3.1, Cor. 3.2] and [17, Thm. 1.1, Thm. 1.2]. □

Remark 2.1.2. It follows from part (2) of the above theorem, together with the usual duality between spaces of modular forms and Hecke algebras, that classical height one primes in T_{N} are in one-to-one correspondence with Galois conjugacy classes of p-ordinary normalized Hecke eigenforms of tame level N and weight k ≥ 2 (defined over an algebraic closure of Q̄_{p}). For such an eigenform f, we denote the corresponding prime ideal by ℘_{f} and for a classical height one prime ideal ℘ in T_{N}, we denote the corresponding eigenform by f_{℘}.

The new quotient T_{N}^{new} of T_{N} is defined to be the quotient of T_{N} that acts faithfully on the space of newforms. The following theorem of Hida summarizes its basic properties.

Theorem 2.1.3.

(1) T_{N}^{new} is a finite and reduced torsion free Λ-algebra.

(2) The classical height one primes of T_{N}^{new} correspond (under pull-back) to those classical height one primes ℘ of T_{N} for which the corresponding normalized eigenform f_{℘} is of tame conductor N.

(3) If ℘ is a classical height one prime ideal of Λ, then T_{N}^{new} ⊗_{Λ} Λ_{℘} is a finite étale extension of the discrete valuation ring Λ_{℘}.

Proof. Parts (1) and (2) follow from the results of [18]. See, in particular, Corollaries 3.3 and 3.7. Part (3) is a rewording of [17, Cor. 1.4]. □

2.2. Galois representations. In this section we recall the basic facts regarding Galois representations attached to Hida families. As above, we fix a tame level N prime to p. Recall that L denotes the field of fractions of Λ.

Theorem 2.2.1. There is a continuous Galois representation

ρ : G_{Q} → GL_{2}(T_{N}^{new} ⊗_{Λ} L),

characterized by the following properties:

(1) ρ is unramified away from Np.

(2) If ℓ is a prime not dividing Np then ρ(Frob_{ℓ}) has characteristic polynomial equal to X^{2} − T_{ℓ}X + ⟨ℓ⟩_{Np}ℓ^{−1} ∈ T_{N}^{new}[X]. (Recall that ⟨ℓ⟩_{Np} denotes the unit in Z_{p}[[Z_{p,N}^{×}]] corresponding to the element ℓ ∈ Z_{p,N}^{×}.)

The representation ρ satisfies the following additional properties:

(3) ρ is absolutely irreducible.

(4) The determinant of ρ is equal to the following character:

G_{Q} → Ẑ^{×} → Z_{p,N}^{×} ⟶^{x↦⟨x⟩_{Np}x_{p}^{−1}} Z_{p}[[Z_{p,N}^{×}]]^{×} ⊂ (T_{N}^{new})^{×}.

(Here the first arrow is the full cyclotomic character, and x_{p} denotes the image of x under the projection Z_{p,N}^{×} → Z_{p}^{×}.)

(5) The space of I_{p}-coinvariants of ρ is free of rank one over T_{N}^{new} ⊗_{Λ} L and Frob_{p} acts on this space through the eigenvalue U_{p} ∈ T_{N}^{new}.

Proof. See [17, Thm. 2.1] and [14, Thm. 2.6]. □

Remark 2.2.2. The representation ρ encodes all of the representations attached to the classical modular forms occurring in the Hida family determined by T_{N}^{new}. Indeed, if T̂_{N}^{new} denotes the normalization of T_{N}^{new}, then we may descend ρ to a two dimensional representation defined over T̂_{N}^{new} (see [18, Theorem 2.1]). If ℘ is a classical height one prime ideal in T_{N}^{new}, then we have an isomorphism (T_{N}^{new})_{℘} ≅ (T̂_{N}^{new})_{℘}, by part (3) of Theorem 2.1.3, and consequently we may descend ρ to a two dimensional representation over (T_{N}^{new})_{℘}. Reducing this representation modulo the maximal ideal of this local ring, we obtain a representation ρ̄_{℘} : G_{Q} → GL_{2}(O(℘)[1/p]). Part (2) of Theorem 2.2.1 shows that this is the usual absolutely irreducible two dimensional Galois representation attached to the newform f_{℘}.

From the Galois representation ρ, we may also construct various related Galois representations corresponding to the minimal and maximal primes of T_{N}^{new}.

We consider first the case of minimal primes. Note that T_{N}^{new} ⊗_{Λ} L ≅ ∏_{𝔞}(T_{N}^{new})_{𝔞} where the product is taken over all minimal primes of T_{N}^{new}. For 𝔞 a fixed minimal prime ideal of T_{N}^{new}, let ρ_{𝔞} denote the representation

ρ_{𝔞} : G_{Q} → GL_{2}((T_{N}^{new})_{𝔞})

obtained by composing ρ with the projection T_{N}^{new} ⊗_{Λ} L → (T_{N}^{new})_{𝔞}.

Since ρ_{𝔞} takes values in a field, we may define its tame conductor by the usual formula. That is, if ρ_{𝔞} acts on the two dimensional (T_{N}^{new})_{𝔞}-vector space V, then the exponent of a prime ℓ ≠ p in the tame conductor is equal to

dim V − dim V_{I_{ℓ}} + ∫_{0}^{∞} (dim V − dim V_{I_{ℓ}^{u}}) du,

where {I_{ℓ}^{u}}_{u≥0} denotes the usual filtration of I_{ℓ} by higher ramification groups, indexed by the upper numbering. (Note that this formula is often expressed in terms of invariants under the ramification groups, rather than coinvariants. One obtains the same value with either formulation, since dim V_{I_{ℓ}^{u}} = dim V^{I_{ℓ}^{u}} for any value of u.)

Proposition 2.2.3. If 𝔞 is any minimal prime of T_{N}^{new}, then the tame conductor of ρ_{𝔞} is equal to N.

Proof. Write T̂ to denote the normalization of T_{N}^{new}/𝔞; this is the component of the normalization of T_{N}^{new} cut out by the minimal prime 𝔞. Since it is Cohen-Macaulay (being normal and of dimension two) and finite over Λ, it is in fact finite flat over Λ by [24, Thm. 23.1].

Let V denote a two dimensional vector space over the fraction field of T_{N}^{new}/𝔞 on which ρ_{𝔞} acts and let M be a free rank two T̂-lattice in V invariant under G_{Q}. If ℘ is a classical height one prime of T_{N}^{new} containing 𝔞, then as was observed above, the injection

(T_{N}^{new}/𝔞)_{℘} → T̂_{℘}

is an isomorphism. Moreover, G_{Q} acts on the O(℘)[1/p]-vector space V (℘) := (M/℘M)[1/p] via the usual Galois representations ρ̄_{℘} attached to newform f_{℘} of tame conductor N.

The ratio of the tame conductor of ρ_{𝔞} and the tame conductor of ρ̄_{℘} (that is, N) is equal to

(2.1)  ∏_{ℓ≠p} ℓ^{dim V (℘)_{I_{ℓ}} −dim V_{I_{ℓ}}};

furthermore, each of the exponents appearing in this expression is non-negative (i.e. dim V (℘)_{I_{ℓ}} bounds dim V_{I_{ℓ}} from above). (See [23, §1]). Thus to prove the proposition, we must show that dim V (℘)_{I_{ℓ}} = dim V_{I_{ℓ}} for each prime ℓ ≠ p.

If dim V (℘)_{I_{ℓ}} = 2, then ℓ does not divide N. In this case, ρ_{𝔞} is unramified at ℓ, and so dim V_{I_{ℓ}} = 2 as well. Conversely, if dim V_{I_{ℓ}} = 2, then the same is true of dim V (℘)_{I_{ℓ}} (since the latter dimension bounds the former dimension from above). If dim V (℘)_{I_{ℓ}} = 0, then also dim V_{I_{ℓ}} = 0 (again, since the latter dimension is bounded above by the former). Thus it remains to show that if dim V (℘)_{I_{ℓ}} > 0, then dim V_{I_{ℓ}} > 0. This we now do.

Note that the formula (2.1) (and the fact that the tame conductor of V (℘) is equal to N, and thus is independent of the particular choice of ℘ containing 𝔞) shows that dim V (℘)_{I_{ℓ}} is independent of the choice of the classical height one prime ℘ containing 𝔞. Thus we may assume that dim V (℘)_{I_{ℓ}} is positive for every classical height one prime ℘ containing 𝔞. There is a natural map I_{ℓ} → Z_{p}(1), given by projection onto the p-Sylow subgroup of the tame quotient of I_{ℓ}. Let J_{ℓ} denote the kernel of this projection. Since J_{ℓ} has order prime to p, it acts on M through a finite quotient of order prime to p and so certainly dim V_{J_{ℓ}} = dim V (℘)_{J_{ℓ}} for any classical height one prime ℘. Let σ denote a topological generator of Z_{p}(1) and consider the matrix ρ_{𝔞}(σ) − I acting on V_{J_{ℓ}}. (Here I denotes the identity matrix.) By assumption, the determinant of this matrix (which lies in T̂) lies in ℘T̂ for every classical height one prime ℘ of T̂. Thus it lies in ℘′T̂ for every classical height one prime ℘′ of Λ. (Recall that these primes are unramified in T̂, by part (3) of Theorem 2.1.3.) Since T̂ is finite flat over Λ, we see that this determinant vanishes and thus ρ_{𝔞}(σ) admits a non-zero coinvariant quotient of V_{J_{ℓ}}. This proves that dim V_{I_{ℓ}} > 0 as required. □

As a byproduct of the proof of the preceding proposition, we also obtain the following useful result.

Proposition 2.2.4. Let ℘ be a classical height one prime ideal in T_{N}^{new} and let L denote a choice of two dimensional (T_{N}^{new})_{℘}-lattice on which ρ acts. (In particular, L/℘L is a two dimensional O(℘)[1/p]-vector space on which ρ̄_{℘} acts.) If ℓ is any prime distinct from p, then L_{I_{ℓ}} is a free (T_{N}^{new})_{℘}-module and there are natural isomorphisms L_{I_{ℓ}} ⊗_{Λ} L → (L ⊗_{Λ} L)_{I_{ℓ}} and L_{I_{ℓ}}/℘L_{I_{ℓ}} → (L/℘L)_{I_{ℓ}}.

Proof. We let 𝔞 denote the minimal ideal contained in ℘ (unique since (T_{N}^{new})_{℘} is a discrete valuation ring) and employ the notation introduced in the proof of the preceding proposition. The lattice L is a free rank two (T_{N}^{new})_{℘}-module, which we may regard as being embedded in a G_{Q}-equivariant fashion in V.

For general reasons, the composite surjection L → L/℘L → (L/℘L)_{I_{ℓ}} induces an isomorphism L_{I_{ℓ}}/℘L_{I_{ℓ}} ≅ (L/℘L)_{I_{ℓ}}. Thus a minimal set of generators for L_{I_{ℓ}} as a (T_{N}^{new})_{℘}-module contains at most dim(L/℘L)_{I_{ℓ}} elements. On the other hand the surjection V → V_{I_{ℓ}} induces a surjection L_{I_{ℓ}} ⊗_{Λ} L → V_{I_{ℓ}} and hence L_{I_{ℓ}} ⊗_{Λ} L is of dimension at least dim V_{I_{ℓ}}. The proof of Proposition 2.2.3 shows that dim(L/℘L)_{I_{ℓ}} = dim V_{I_{ℓ}} and hence (taking into account the fact that (T_{N}^{new})_{℘} is a discrete valuation ring) we may conclude that L_{I_{ℓ}} is free over (T_{N}^{new})_{℘} of rank equal to dim V_{I_{ℓ}}. Thus the surjection L_{I_{ℓ}} ⊗_{Λ} L → V_{I_{ℓ}} is an isomorphism as claimed. □

We now turn to the Galois representations attached to maximal ideals. If 𝔪 is a maximal ideal of T_{N}^{new}, then the localization (T_{N}^{new})_{𝔪} is a direct factor of T_{N}^{new} and so (T_{N}^{new})_{𝔪}⊗_{Λ}L is a direct factor of T_{N}^{new}⊗_{Λ}L. We let ρ_{𝔪} denote the representation

ρ_{𝔪} : G_{Q} → GL_{2}((T_{N}^{new})_{𝔪} ⊗_{Λ} L)

obtained by composing ρ with the projection T_{N}^{new} ⊗_{Λ} L → (T_{N}^{new})_{𝔪} ⊗_{Λ} L.

The next result describes the residual representation ρ̄_{𝔪} attached to a maximal ideal of T_{N}^{new}.

Theorem 2.2.5. If 𝔪 is a maximal ideal of T_{N}^{new}, then attached to 𝔪 is a semisimple representation ρ̄_{𝔪} : G_{Q} → GL_{2}(T_{N}^{new}/𝔪), uniquely determined by the properties:

(1) ρ̄_{𝔪} is unramified away from Np.

(2) If ℓ is a prime not dividing Np then ρ̄_{𝔪}(Frob_{ℓ}) has characteristic polynomial equal to X^{2} − T_{ℓ}X + ⟨ℓ⟩_{Np}ℓ^{−1} mod 𝔪 ∈ (T_{N}^{new}/𝔪)[X].

Furthermore, ρ̄_{𝔪} satisfies the following condition:

(3) The restriction of ρ̄_{𝔪} to G_{p} has the following shape (with respect to a suitable choice of basis):

(\begin{matrix}χ & ∗ \\ 0 & ψ \end{matrix}) ,

where χ and ψ are (T_{N}^{new}/𝔪)^{×}-valued characters of G_{p} such that ψ is un-

ramified and ψ(Frob_{p}) = U_{p} mod 𝔪.

Proof. The representation ρ̄_{𝔪} is constructed in the usual way, by choosing an integral model for ρ over T̂_{N}^{new}, reducing this model modulo a maximal ideal 𝔪̂ lifting 𝔪, semi-simplifying and then descending (if necessary) from T̂_{N}^{new}/𝔪̂ to T_{N}^{new}/𝔪. The stated properties follow from the corresponding properties of ρ. □

Proposition 2.2.6. If 𝔪 is a maximal ideal of T_{N}^{new} for which the associated residual representation ρ̄_{𝔪} is irreducible, then ρ_{𝔪} admits a model over (T_{N}^{new})_{𝔪} (which we denote by the same symbol)

ρ_{𝔪} : G_{Q} → GL_{2}((T_{N}^{new})_{𝔪}),

unique up to isomorphism.

Proof. This follows from the irreducibility of ρ̄_{𝔪} and the fact that the traces of ρ_{𝔪} lie in (T_{N}^{new})_{𝔪}. (See [2].) □

Definition 2.2.7. If k is a field of characteristic p and ρ̄: G_{Q} → GL_{2}(k) is a continuous representation for which ρ̄_{|G_{p}} is reducible, then we say that ρ̄ is p-distinguished if the semisimplification of ρ̄_{|G_{p}} is non-scalar (i.e. if the two characters appearing as Jordan-Hölder factors of the reducible representation ρ̄_{|G_{p}} are distinct).

Let ⟨ε⟩_{p} : G_{Q} → Z_{p}[[Z_{p}^{×}]]^{×} denote the character obtained by composing the p-adic cyclotomic character ε : G_{Q} → Z_{p}^{×} with the map ⟨ ⟩_{p} : Z_{p}^{×} → Z_{p}[[Z_{p}^{×}]]^{×}.

Proposition 2.2.8. Let 𝔪 be a maximal ideal of T_{N}^{new} for which the associated residual representation ρ̄_{𝔪} is irreducible and let L denote a choice of a free rank two (T_{N}^{new})_{𝔪}-module on which the representation ρ_{𝔪} acts. (Such an L exists by Proposition 2.2.6.) Let L_{0} denote the maximal submodule of L on which I_{p} acts through the character ⟨ε⟩_{p}ε^{−1}. If ρ̄_{𝔪} is p-distinguished, then each of L_{0} and L/L_{0} is free of rank one over (T_{N}^{new})_{𝔪} and the G_{p}-action on L/L_{0} is unramified.

Proof. Theorem 2.2.1 shows that L_{0} ⊗_{Λ} L is a one-dimensional direct summand of L⊗_{Λ}L and that G_{p} acts on this space and on (L⊗_{Λ}L)/(L_{0}⊗_{Λ}L) through a character. Thus G_{p} acts on each of L_{0} and L/L_{0} through a character. By assumption, L/𝔪L is a p-distinguished representation of G_{p} and so (L/L_{0})/𝔪(L/L_{0}) must be onedimensional over T_{N}/𝔪. The proposition now follows by Nakayama’s Lemma. □

Suppose that 𝔪 is a maximal ideal of T_{N}^{new} satisfying the hypothesis of the preceding proposition. As in the statement of that proposition, let L denote a choice of free rank two (T_{N}^{new})_{𝔪}-module on which the representation ρ_{𝔪} acts and let L_{0} be the maximal submodule of L on which I_{p} acts through the character ⟨ε⟩_{p}ε^{−1}. The group G_{Q} then acts on the quotient L/𝔪L via the residual representation ρ̄_{𝔪}.

Since the space L/L_{0} is a free rank one quotient of L on which the G_{p}-action is unramified (by Proposition 2.2.8), we see that (L/L_{0})/𝔪(L/L_{0}) is a one dimensional unramified quotient of L/𝔪L.

Definition 2.2.9. If k is a finite field of characteristic p and ρ̄: G_{Q} → GL_{2}(k) is a continuous Galois representation acting on a two dimensional k-vector space V, then a p-stabilization of ρ̄ is a choice of a one dimensional quotient of V on which the G_{p}-action is unramified.

Definition 2.2.10. The discussion preceding Definition 2.2.9 shows that if 𝔪 is a maximal ideal of T_{N}^{new} for which ρ̄_{𝔪} is irreducible and p-distinguished, then the quotient (L/L_{0})/𝔪(L/L_{0}) (in the notation of that discussion) forms a p-stabilization of ρ̄_{𝔪}. We refer to this as the canonical p-stabilization of ρ̄_{𝔪} attached to the maximal ideal 𝔪.

2.3. The reduced Hida algebra.

Definition 2.3.1. For any level N, we let T′_{N} denote the Λ-subalgebra of T_{N} generated by the Hecke operators T_{ℓ} for ℓ prime to Np, together with the operator U_{p}.

Since T′_{N} is a subalgebra of the finite flat Λ-algebra T_{N}, it is certainly finite and torsion free over Λ. It turns out that T′_{N} is also reduced. (Being reduced is a standard property of Hecke algebras in which we omit the operators indexed by the primes dividing the level.) In fact, we can be somewhat more precise.

If M is a divisor of N, then restricting the action of the prime-to-N Hecke operators to p-ordinary forms of level dividing M yields a surjective map of Λalgebras T′_{N} → T′_{M}. Composing this with the composite T′_{M} ⊂ T_{M} → T_{M}^{new} yields a map

(2.2)  T′_{N} → T_{M}^{new}.

Taking the product of these maps over all divisors M of N, we obtain a map

(2.3)  T′_{N} → ∏_{M|N} T_{M}^{new}.

Proposition 2.3.2.

(1) The map (2.3) is injective and induces an isomorphism after localizing at any classical height one prime ideal of Λ (and hence also after tensoring over Λ with its fraction field L).

(2) If ℘ is a classical height one prime ideal of Λ, then T′_{N} ⊗_{Λ} Λ_{℘} is a finite étale extension of the discrete valuation ring Λ_{℘}.

Proof. Part (1) follows from the theory of newforms for ordinary families developed in [18]. Part (2) then follows from part (1) together with part (3) of Theorem 2.1.3.

□

Taking the product of the Galois representations

G_{Q} → GL_{2}(T_{M}^{new} ⊗_{Λ} L)

given by Theorem 2.2.1, as M ranges over all divisors of N, and taking into account part (1) of the preceding proposition, we obtain a Galois representation

ρ : G_{Q} → GL_{2}(T′_{N} ⊗_{Λ} L)

satisfying the analogue of Theorem 2.2.1.

Just as in the preceding section, we may reduce ρ modulo a height one prime ideal ℘ or a maximal ideal 𝔪 of T′_{N}. We denote the corresponding residual representations by ρ̄_{℘} and ρ̄_{𝔪} respectively. Similarly, we may localize T′_{N} at any maximal ideal 𝔪 and obtain a corresponding representation

ρ_{𝔪} : G_{Q} → GL_{2}((T′_{N})_{𝔪} ⊗_{Λ} L).

If ρ̄_{𝔪} is irreducible, then the analogue of Proposition 2.2.6 holds (by the same appeal to the results of [2]) and so we obtain a uniquely determined representation

ρ_{𝔪} : G_{Q} → GL_{2}((T′_{N})_{𝔪}).

2.4. The reduced Hida algebras attached to ρ̄. If k is a finite field of characteristic p and ρ̄: G_{Q} → GL_{2}(k) is a continuous two dimensional Galois representation defined over k, then we say that ρ̄ is p-ordinary if ρ̄ restricted to G_{p} has an unramified quotient of dimension one over k. Clearly ρ̄ admits a p-stabilization, in the sense of Definition 2.2.9, if and only if ρ̄ is p-ordinary. If ρ̄ is furthermore pdistinguished (in the sense of Definition 2.2.7), then ρ̄ admits at most two choices of p-stabilization (and does admit two such choices precisely when the determinant of ρ̄ is unramified at p).

Let us now fix such a representation ρ̄ : G_{Q} → GL_{2}(k) and let \overline{V} be a two dimensional k-vector space on which ρ̄ acts. We assume that ρ̄ is irreducible, odd, p-ordinary and p-distinguished and we fix a choice of p-stabilization of ρ̄. We assume that k is equal to the field generated by the traces of ρ̄. (If it were not, we could replace k by this field of traces and descend ρ̄ to the smaller field – see [29, Section 6].) Finally, we suppose that ρ̄ is modular (i.e. that it arises as the residual representation attached to a modular form of some weight and level defined over Q̄_{p}). Our goal in this section is to define, for Σ a finite set of primes, the reduced ordinary Hecke algebra T_{Σ}(ρ̄) attached to a modular p-ordinary residual representation and to describe its basic properties.

We let N(ρ̄) denote the tame conductor of ρ̄. If ℓ ≠ p is prime, then define

m_{ℓ} = dim_{k} \overline{V}_{I_{ℓ}}

and for any finite set of primes Σ that does not contain p, write

N(Σ) = N(ρ̄) ∏_{ℓ∈Σ} ℓ^{m_{ℓ}}.

Theorem 2.4.1. There is a unique maximal ideal 𝔪 of T′_{N(Σ)} such that ρ̄_{𝔪}, with its canonical p-stabilization, is isomorphic to ρ̄, with its given p-stabilization.

Proof. The uniqueness is clear. The existence follows from the assumption that ρ̄ is modular and the results of [5]. □

Proposition 2.4.2. If 𝔪 denotes the maximal ideal of T′_{N(Σ)} of the preceding theorem, then there is a unique maximal ideal 𝔫 of T_{N(Σ)} satisfying the following conditions:

(1) 𝔫 lifts 𝔪.

(2) T_{ℓ} ∈ 𝔫 for each ℓ ∈ Σ.

(3) The natural map of localizations (T′_{N(Σ)})_{𝔪} → (T_{N(Σ)})_{𝔫} is an isomorphism of Λ-algebras.

In particular, (T′_{N(Σ)})_{𝔪} is a finite flat Λ-algebra. Also, the image of T_{ℓ} in the localization (T_{N(Σ)})_{𝔫} vanishes for each ℓ ∈ Σ.

Proof. This is a variant of [37, Prop. 2.1.5] and is proved in an analogous manner. The second to last claim follows from the rest of the proposition together with part (1) of Theorem 2.1.1. □

Definition 2.4.3. We let T_{Σ}(ρ̄) (or simply T_{Σ}, if ρ̄ is understood) denote the localization of T′_{N(Σ)} at the maximal ideal whose existence is guaranteed by Theorem 2.4.1. We let ρ_{Σ} : G_{Q} → GL_{2}(T_{Σ}) denote the Galois representation attached to this local factor of T′_{N(Σ)} as discussed in Section 2.3. Recall that ρ_{Σ} is characterized by the following property: If ℓ is a prime not dividing N(Σ)p, then ρ_{Σ}(Frob_{ℓ}) has trace equal to T_{ℓ} ∈ T_{Σ}.

Taking into account Proposition 2.4.2, we see that T_{Σ} is a reduced and finite flat Λ-algebra. Note that if Σ ⊂ Σ′, then N(Σ) | N(Σ′) and the natural surjection T′_{N(Σ′)} → T′_{N(Σ)} induces a surjection T_{Σ′} → T_{Σ}. The Galois representations ρ_{Σ′} and ρ_{Σ} are evidently compatible with this surjection.

We refer to Spec T_{Σ} as the universal p-ordinary family of newforms, or sometimes simply “the Hida family” attached to ρ̄ and our chosen p-stabilization that is minimally ramified outside Σ. Localizing ρ_{Σ} over Spec T_{Σ}, we obtain a two dimensional vector bundle on which G_{Q} acts, which we refer to as the universal family of Galois representations over Spec T_{Σ}. If Σ ⊂ Σ′, then the surjection T_{Σ} → T_{Σ′} induces a closed embedding Spec T_{Σ} → Spec T_{Σ′} and the universal family of Galois representations on the target pulls back to the universal family of Galois representations on the source. If we consider all Σ simultaneously, then we obtain an Ind-scheme

“lim_{\overrightarrow{Σ}}” Spec T_{Σ}

and a family of two dimensional Galois representations lying over it. We refer to this Ind-scheme as the universal p-ordinary family of newforms, or simply “the Hida family” attached to ρ̄ and its chosen p-stabilization.

The Hida family corresponding to Σ = ∅ will play a special role; we refer to it as the minimal Hida family attached to ρ̄ and our chosen p-stabilization.

When ρ̄ is irreducible after restriction to the quadratic field of discriminant (−1)^{(p−1)/2}p, the results of Wiles and Taylor–Wiles [37, 34], as strengthened by Diamond [6], in fact allow us to identify the rings T_{Σ}, and their accompanying Galois representations ρ_{Σ}, with certain universal deformation rings, and their accompanying universal Galois representations, attached to the residual representation ρ̄. Specifically, let R_{Σ} denote the universal deformation ring parameterizing lifts of ρ̄ which are p-ordinary and whose tame conductor coincides with that of ρ̄ at primes not in Σ ∪ {p}. The representation ρ_{Σ} induces a map R_{Σ} → T_{Σ}, which by [6] is an isomorphism after tensoring with the quotient of Λ by any classical height one prime; it follows that in fact R_{Σ} → T_{Σ} is an isomorphism, as claimed.

2.5. Branches. We now prove some results concerning the irreducible components of the Hida family attached to ρ̄.

Definition 2.5.1. If 𝔞 is a minimal prime ideal in T_{Σ} for any Σ as above, then we write T(𝔞) := T_{Σ}/𝔞. Note that T(𝔞) is a local domain, finite over Λ. We write K(𝔞) to denote the fraction field of T(𝔞). (Thus there is an isomorphism K(𝔞) ≅ T(𝔞) ⊗_{Λ} L.) We let ρ(𝔞) denote the Galois representation

ρ(𝔞) : G_{Q} → GL_{2}(T(𝔞))

induced by ρ_{Σ}.

Proposition 2.5.2. If 𝔞 is a minimal prime of T_{Σ} for any Σ as above, then there is a unique divisor N(𝔞) of N(Σ) and a unique minimal prime 𝔞′ ⊆ T_{N(𝔞)}^{new} sitting over 𝔞 such that

\xymatrix{T_{Σ} \ar[r] \ar[d] & T′_{N(Σ)} \ar[r] & ∏_{M|N(Σ)} T_{N(𝔞)}^{new} \ar[d] \\ T_{Σ}/𝔞 \ar[r]^{=} & T(𝔞) \ar[r] & T_{N(𝔞)}^{new}/𝔞′}

commutes.

Proof. Since T_{Σ} is finite over Λ, the minimal primes of T_{Σ} are in bijection with the local components of T_{Σ} ⊗_{Λ} L. Since T_{Σ} is a local factor of T′_{N(Σ)}, these local components are included in the local components of T′_{N(Σ)} ⊗_{Λ} L. By part (1) of Proposition 2.3.2, we have that

T′_{N(Σ)} ⊗_{Λ} L ≅ ∏_{M|N(Σ)} T_{M}^{new} ⊗_{Λ} L.

The local components of ∏_{M|N(Σ)} T_{M}^{new} ⊗_{Λ}L are in one-to-one correspondence with its minimal primes. Thus our given minimal prime 𝔞 gives rise to a minimal prime of this ring. However, any such minimal prime corresponds to a minimal prime 𝔞′ in T_{N(𝔞)}^{new} for some N(𝔞) | N(Σ), which establishes the proposition. □

Definition 2.5.3. For a minimal prime ideal 𝔞 of T_{Σ}, we refer to N(𝔞) (of the preceding proposition) as the tame conductor attached to 𝔞 (or to the irreducible component of Spec T_{Σ} corresponding to 𝔞). Moreover, set T(𝔞)^{∘} := T_{N(𝔞)}^{new}/𝔞′.

Remark 2.5.4. Proposition 2.5.2 gives rise to an embedding of local domains T(𝔞) → T(𝔞)^{∘}. (Note that T(𝔞)^{∘} is local since it is a complete finite Λ-algebra and hence a product of local rings. Being a domain, it must be local.)

We next observe that at the generic point of the 𝔞-component of Spec T_{Σ}, the universal Galois representation has conductor equal to the conductor of 𝔞.

Corollary 2.5.5. The tame conductor of the Galois representation ρ(𝔞) equals N(𝔞). (Here we define the tame conductor by regarding ρ(𝔞) as a Galois representation defined over the field of fractions K(𝔞) of T(𝔞).)

Proof. This follows from Propositions 2.2.3 and 2.5.2. □

The next result deals with the classical height one primes in T_{Σ}.

Proposition 2.5.6. Let ℘ be any classical height one prime ideal of T_{Σ}.

(1) The ring T_{Σ} is étale over Λ (and so regular) in a neighborhood of ℘; consequently, ℘ contains a unique minimal prime 𝔞 of T_{Σ} and the natural map of localizations (T_{Σ})_{℘} → T(𝔞)_{℘} is an isomorphism.

(2) Thinking of ℘ as a height one prime of T(𝔞), the map T(𝔞) → T(𝔞)^{∘} is an isomorphism in a neighborhood of ℘. Consequently, there is a unique height one prime ℘′ of T(𝔞)^{∘} that pulls back to ℘ under this map and the map of localizations T(𝔞)_{℘} → T(𝔞)_{℘′}^{∘} is an isomorphism.

Proof. Both claims follow directly from Proposition 2.3.2. □

In the situation of the preceding proposition, we write O(℘)^{∘} := O(℘′); this is a finite extension of O(℘). Recall that we have defined the classical newform f_{℘′} attached to the height one prime ideal ℘′ of T(𝔞)^{∘} (thought of as a classical height one prime of T_{N(𝔞)}^{new}). We write f_{℘} := f_{℘′}; part (2) of Theorem 2.1.3 implies that f_{℘} lies in S_{k}(N(𝔞)p^{∞}, O(℘)^{∘})_{new}^{ord}[κ_{℘}].

2.6. Hecke eigenvalues. In this section, we will give two lemmas that describe Hecke eigenvalues at primes dividing the level in terms of the corresponding Galois representation. The first lemma treats the case of a single modular form and the second discusses the case of a Hida family.

Let f denote a classical newform over Q̄_{p} of some weight and of tame level N. Let ρ_{f} denote the p-adic Galois representation attached to f, let V_{f} denote its underlying space and let a_{ℓ}(f) denote its Hecke eigenvalues. If ℓ does not divide N, then ρ_{f} is unramified at ℓ and a_{ℓ}(f) is equal to the trace of ρ_{f}(Frob_{ℓ}). The following lemma describes a_{ℓ}(f) for ℓ dividing N in terms of ρ_{f}.

Lemma 2.6.1. If ℓ is a prime dividing N, then the following are equivalent: (1) a_{ℓ}(f) is a non-unit.

(2) a_{ℓ}(f) = 0.
(3) V_{I_{ℓ}} = 0.

If these equivalent conditions do not hold, then we have that V_{I_{ℓ}} is one dimensional and a_{ℓ}(f) is equal to the eigenvalue of Frob_{ℓ} acting on this line.

Proof. This is standard. □

Fix a tame level N and a maximal ideal 𝔫 in T_{N}^{new}. Let V denote the free rank two T_{N}^{new} ⊗_{Λ} L-module on which ρ_{𝔫} acts. The following lemma describes T_{ℓ} for ℓ dividing N in terms of ρ_{𝔫}.

Lemma 2.6.2. If ℓ is a prime dividing N, then the following are equivalent: (1) T_{ℓ} lies in the maximal ideal 𝔫.

(2) T_{ℓ} ≡ 0 mod ℘ for some classical height one prime ideal ℘ of (T_{N}^{new})_{𝔫}.

(3) T_{ℓ} ≡ 0 mod ℘ for every classical height one prime ideal ℘ of (T_{N}^{new})_{𝔫}.

(4) T_{ℓ} = 0 in (T_{N}^{new})_{𝔫}.

(5) V_{I_{ℓ}} = 0. If these equivalent conditions do not hold, then we have that V_{I_{ℓ}} is free of rank one over (T_{N}^{new})_{𝔫} ⊗_{Λ} L and T_{ℓ} is equal to the eigenvalue of Frob_{ℓ} acting on this free rank one module.

Proof. It is clear that (4) ⟹ (3) ⟹ (2) ⟹ (1). Also, Condition (1) implies that a_{ℓ}(f_{℘}) is contained in the maximal ideal of O(f_{℘}) = O(℘) for each classical height one prime ℘ of (T_{N}^{new})_{𝔫}. By Lemma 2.6.1, a_{ℓ}(f_{℘}) = 0 for each such prime and thus (1) implies (3). Proposition 2.2.4, together with Lemma 2.6.1, shows that (3) and (5) are equivalent. Lastly, Lemma 2.6.3 (below), together with Theorem 2.1.3, establishes that (3) implies (4).

Suppose now that these equivalent conditions do not hold. Then, Proposition 2.2.4 and Lemma 2.6.1 show that the space V_{I_{ℓ}} is free of rank one over (T_{N}^{new})_{𝔫} ⊗_{Λ} L as claimed. Now let A_{ℓ} ∈ (T_{N}^{new})_{𝔫} ⊗_{Λ} L denote the eigenvalue of Frob_{ℓ} acting on this module. By Proposition 2.2.4, A_{ℓ} ∈ (T_{N}^{new})_{℘} for each classical height one prime ℘ of T_{N}^{new}. By Lemma 2.6.1, we see that T_{ℓ} − A_{ℓ} lies in the maximal ideal of this ring for each such prime ℘. Then, by Lemma 2.6.3 (below) and Theorem 2.1.3, T_{ℓ} = A_{ℓ}. □

Lemma 2.6.3. Let T be a reduced finite free Λ-algebra such that T ⊗_{Λ} Λ_{℘} is an étale extension of Λ_{℘} for every classical height one prime ℘ of Λ. If x ∈ T ⊗ L lies in the maximal ideal of T_{℘′} for every classical height one prime ℘′ of T, then x = 0.

Proof. Let ℘ be a classical height one prime of Λ. Since T ⊗_{Λ} Λ_{℘} is étale over Λ, we see that ℘T ⊗_{Λ} Λ_{℘} = ⋂_{℘′} ℘′T_{℘′}, where the intersection runs over all height one primes ℘′ of T lying over ℘. Our assumption on x thus implies that x ∈ ℘T ⊗_{Λ} Λ_{℘} for all classical height one primes ℘ of Λ. As T is free over Λ, the lemma follows once we note that ⋂_{℘} ℘Λ_{℘} = 0 (the intersection taking place in L). Indeed, the numerator of any element in this intersection lies in an infinite number of height one primes of Λ and so vanishes. □

2.7. Λ-adic modular forms and Euler factors. For each minimal prime 𝔞 of T_{Σ}, we can define a formal q-expansion along the partial normalization Spec T(𝔞)^{∘} of the component Spec T(𝔞) of Spec T_{Σ} that interpolates the q-expansions of the newforms f_{℘} obtained from the classical primes ℘ in Spec T(𝔞). Namely, if we write T (𝔞)^{∘} = T_{N(𝔞)}^{new}/𝔞′, as in Definition 2.5.3, then we define f(𝔞, q) ∈ T(𝔞)^{∘}[[q]] via

f(𝔞, q) = ∑_{n≥1}(T_{n} mod 𝔞′)q^{n}.

(Here we have written T_{n} rather than U_{p}^{r}T_{n′}, when n is of the form n = n′p^{r} with (n′, p) = 1, for the sake of uniformity of notation.)

An alternative way of describing this formal q-expansion along Spec T(𝔞)^{∘} is to describe the corresponding Euler factors.

Definition 2.7.1. Let 𝔞 be a minimal prime of T_{Σ}. As above, write T(𝔞)^{∘} ≅ T_{N(𝔞)}^{new}/𝔞′, For each prime ℓ ≠ p, define the reciprocal Euler factor E_{ℓ}(𝔞, X) ∈ T(𝔞)^{∘}[X] via the usual formula:

E_{ℓ}(𝔞, X) := \begin{cases}1 − (T_{ℓ} mod 𝔞′)X + ⟨ℓ⟩_{N(𝔞)p}ℓ^{−1}X^{2} & if ℓ is prime to N(𝔞) \\ 1 − (T_{ℓ} mod 𝔞′)X & otherwise. \end{cases}

For the sake of completeness, define E_{p}(𝔞, X) = (1 − (U_{p} mod 𝔞′)X).

In terms of these reciprocal Euler factors, the formal q-expansion f(𝔞, q) is characterized by the fact that its “formal Mellin transform” is equal to the formal Dirichlet series

∏_{ℓ} E_{ℓ}(𝔞, ℓ^{−s})^{−1}.

We will now give a Galois theoretic description of these reciprocal Euler factors.

Proposition 2.7.2. Let 𝔞 be a minimal prime of T_{Σ} and let V denote a two dimensional K(𝔞)-vector space on which the Galois representation ρ(𝔞) acts. If ℓ ≠ p is prime, then the Euler factor E_{ℓ}(𝔞, X) ∈ K(𝔞)[X] is equal to the determinant det(Id − Frob_{ℓ} X | V_{I_{ℓ}}).

Proof. This follows from Lemma 2.6.2. □

Since T(𝔞)^{∘} ⊂ T(𝔞)[1/p], we may in particular regard the reciprocal Euler factors E_{ℓ}(𝔞, X) and the formal q-expansion f(𝔞, q) as varying over Spec T(𝔞)[1/p]. Let us close this section by signaling a phenomena which will be fundamental to all that follows: the reciprocal Euler factors E_{ℓ}(𝔞, X) (or equivalently the formal q-expansions f(𝔞, q)) do not extend in a well-defined fashion over Spec T_{Σ}[1/p]. In general, if ℘ is a (necessarily non-classical) height one prime lying in the intersection of two different components of Spec T_{Σ}[1/p], say Spec T(𝔞_{1})[1/p] and Spec T(𝔞_{2})[1/p], then E_{ℓ}(𝔞_{1}, X) and E_{ℓ}(𝔞_{2}, X) (and hence f(𝔞_{1}, q) and f(𝔞_{2}, q)) may have different specializations in O(℘)[1/p][X] (resp. O(℘)[1/p][[q]]) for certain values of ℓ dividing N(𝔞_{1}) or N(𝔞_{2}).

3. Two-variable p-adic L-functions

3.1. Canonical periods. For any level M, we let X_{1}(M) denote the closed modular curve of level Γ_{1}(M) and let C_{1}(M) denote its set of cusps. Let L_{k}(R) denote the set of polynomials of degree less than or equal to k − 2 with coefficients in R and let L̃_{k}(R) denote the associated local system on X_{1}(M).

For N prime to p, let T_{N,r,k}^{∗} (resp. T_{N,r,k}) denote the Hecke algebra corresponding to p-ordinary modular forms (resp. cusp forms) on Γ_{1}(Np^{r}) of weight k. To ease notation, we write

H_{1}(Np^{r}; L̃_{k}(Z_{p})) := H_{1}(X_{1}(Np^{r}); L̃_{k}(Z_{p}))

and

H_{1}(Np^{r}, {cusps}; L̃_{k}(Z_{p})) := H_{1}(X_{1}(Np^{r}); C_{1}(Np^{r}); L̃_{k}(Z_{p})).

We then have a short exact sequence of T_{N,r,k}^{∗}-modules:

0 → H_{1}(Np^{r}; L̃_{k}(Z_{p})) → H_{1}(Np^{r}, {cusps}; L̃_{k}(Z_{p})) → H̃_{0}(C_{1}(Np^{r}); L̃_{k}(Z_{p})) → 0.

(This is a part of the relative homology sequence of the pair (X_{1}(Np^{r}), C_{1}(Np^{r})); the tilde over the H_{0} denotes reduced homology.)

If we localize the above sequence at a maximal ideal 𝔪 in T_{N,r,k}^{∗} corresponding to an irreducible residual G_{Q}-representation, then the localization of H̃_{0}(C_{1}(Np^{r}); Z_{p}) will vanish and the two H_{1} terms will become isomorphic. Passing to ordinary parts (an exact functor) yields an isomorphism

(3.1) H_{1}(Np^{r}; L̃_{k}(Z_{p}))_{𝔪}^{ord} ≅ H_{1}(Np^{r}, {cusps}; L̃_{k}(Z_{p}))_{𝔪}^{ord}

of (T_{N,r,k})_{𝔪} ≅ (T_{N,r,k}^{∗})_{𝔪}-modules. We write (𝔐_{N,r,k})_{𝔪} to denote these isomorphic modules.

There is an action of complex conjugation on X_{1}(Np^{r}) which induces an action on these homology group. We will use a superscript of ± to indicate the ±-eigenspaces for this action; since p is odd, we have (𝔐_{N,r,k})_{𝔪} = (𝔐_{N,r,k})_{𝔪}^{+} ⊕ (𝔐_{N,r,k})_{𝔪}^{−}.

The following proposition is fundamental to the construction of the canonical period.

Proposition 3.1.1. Let 𝔪 be a maximal ideal of T_{N,r,k} whose residual representation is irreducible and p-distinguished. Then (𝔐_{N,r,k})_{𝔪}^{±} is a free (T_{N,r,k})_{𝔪}-module of rank 1. In particular, there exists an isomorphism of (T_{N,r,k})_{𝔪}-modules

(𝔐_{N,r,k})_{𝔪}^{±} ≅^{α_{N,r,k}} (T_{N,r,k})_{𝔪} (3.2)

(although α_{N,r,k} is only well-defined up to a p-adic unit).

Proof. We consider first the case k = 2 and r = 1. Set 𝔐 := 𝔐_{N,1,2} and T := T_{N,1,2}. Let J_{1}(Np) denote the Jacobian of X_{1}(Np). By [37, Thm. 2.1], the ordinary 𝔪-torsion J_{1}(Np)(Q̄)[𝔪]^{ord} is free of rank 2 over T/𝔪. Since

J_{1}(Np)(Q̄)[𝔪] ≅ H^{1}(Np; F_{p})[𝔪],

by dualizing, we have that (𝔐_{N})_{𝔪}⊗T/𝔪 is free of rank 2 over T/𝔪. As (𝔐_{N})_{𝔪}⊗Q_{p} is free of rank 2 over T_{𝔪} ⊗ Q_{p}, it follows that (𝔐_{N})_{𝔪} is free of rank 2 over T_{𝔪}. Passing to ±-subspaces establishes the proposition for k = 2.

The case k > 2 or r > 1 can be deduced from the case k = 2 and r = 1 by Hida theory. We will give the details in Section 3.3 immediately after Proposition 3.3.1.

□

Fix a p-ordinary and p-distinguished eigenform f defined over Q̄_{p} of weight k on Γ_{1}(Np^{r}) whose corresponding maximal ideal in T_{N,r,k} is 𝔪. By the duality between spaces of cusp forms and Hecke algebras, the eigenform f gives rise to a morphism of Z_{p}-algebras

(3.3)  (T_{N,r,k})_{𝔪} → O(f);

here O(f) is the finite extension of Z_{p} in Q̄_{p} generated by the Fourier coefficients of f. Composing (3.3) with the fixed map α_{N,r,k} of (3.2) then yields an O(f)-valued functional

H_{1}(Np^{r}; L̃_{k}(O(f)))_{𝔪}^{±} →^{δ_{f}^{±}} O(f).

On the other hand, integration yields a map

H_{1}(Np^{r}; L̃_{k}(C))^{±} ⟶^{ω_{f}^{±}} C.

If we fix an isomorphism ı : Q̄_{p} ≅ C, then this integration map factors through H_{1}(Np^{r}; L̃_{k}(C))_{𝔪′}^{±}, where 𝔪′ is the maximal ideal in T_{N,r,k} ⊗_{Z_{p},ı} C obtained as the kernel of the C-algebra map

T_{N,r,k} ⊗_{Z_{p},ı} C → C

induced by composing (3.3) with ı. By the multiplicity one theorem, the functionals δ_{f}^{±} (extended to C) and ω_{f}^{±} differ by a non-zero constant and we have

ω_{f}^{±} = Ω_{f}^{±} · δ_{f}^{±}

with Ω_{f}^{±} ∈ C^{×}. The constant Ω_{f}^{±} is the canonical period of f. It is well-defined up to a p-adic unit. (The ambiguity arises from the choice of the map α_{N,r,k} in (3.2)).

Remark 3.1.2. The constructions given here are dual (but equivalent) to the ones given by Vatsal in [35], since here we are working with homology while Vatsal used cohomology.

3.2. One variable p-adic L-functions. The advantage of describing (𝔐_{N,r,k})_{𝔪} as a relative homology group is that the corresponding relative homology classes admit a description via modular symbols. More precisely, for fixed r, there is a map

P^{1}(Q) → H_{1}(Np^{r}, {cusps}; L̃_{k}(Z_{p}))

defined by sending the element a ∈ P^{1}(Q) to the homology class corresponding to the image of the path [∞, a] on the modular curve X_{1}(Np). By projecting to ordinary parts and localizing at 𝔪, we obtain a map

P^{1}(Q) → (𝔐_{N,r,k})_{𝔪},

which we denote by a ↦ {∞, a}. This map allows us to define an (𝔐_{N,r,k})_{𝔪}-valued measure on Z_{p}^{×} in the usual way.

Definition 3.2.1. For any open subset a + p^{r}Z_{p} of Z_{p}^{×}, we define

μ_{k}(a + p^{r}Z_{p}) = U_{p}^{−r}{∞, a/p^{r}} ∈ (𝔐_{N,r,k})_{𝔪}.

Proposition 3.2.2. The function μ_{k} is a measure (i.e. it is additive).

Proof. This is standard. □

For f an eigenform of weight k on Γ_{1}(Np^{r}), let ℘_{f} denote the corresponding height one prime ideal of T_{N,r,k} and let 𝔪 be the unique maximal ideal containing ℘_{f}. By decomposing μ_{k} into its ±-parts and applying the map α_{N,r,k} of (3.2), we may view μ_{k}^{±} as taking values in T_{N,r,k}. Thus μ_{k}^{±} mod ℘_{f} is O(f)-valued and moreover, it is precisely the p-adic L-function attached to f computed with respect to the canonical period Ω_{f}^{±}. That is, μ_{k}^{±} mod ℘_{f} interpolates the special values L(f, χ, 1)/Ω_{f}^{±} where χ is a finite order character of Z_{p}^{×} such that χ(−1) = ±1.

3.3. Canonical periods in families. In [35], it is proven that if f and g are p-ordinary eigenforms of weight k whose residual representations are isomorphic, then their respective p-adic L-functions are congruent. More precisely, it is proven that one can simultaneously choose canonical periods Ω_{f} and Ω_{g} so that the p-adic L-functions computed with respect to these periods are congruent.

We will now describe how one can simultaneously choose canonical periods for all modular forms in a Hida family so that the associated single variable p-adic L-functions vary p-adic analytically in k. These constructions are originally due to Mazur [28] and Kitigawa [22]. Using results of Wiles in [37], we have extended these constructions to a wider class of modular forms.

Let T_{N} be the universal ordinary Hecke algebra defined in Section 2 and let T_{N}^{∗} be the corresponding Hecke algebra defined with respect to all ordinary modular forms rather than just cusp forms. Let 𝔪 ⊆ T_{N}^{∗} be a maximal ideal whose residual representation is irreducible.

For each r ≥ 0, we have an isomorphism

H_{1}(Np^{r}; Z_{p})_{𝔪}^{ord} ≅ H_{1}(Np^{r}, {cusps}; Z_{p})_{𝔪}^{ord}

and passing to the projective limit in r, we obtain a corresponding isomorphism of (T_{N})_{𝔪} ≅ (T_{N})_{𝔪}^{∗}-modules

lim_{\overleftarrow{r}} H_{1}(Np^{r}; Z_{p})_{𝔪}^{ord} ≅ lim_{\overleftarrow{r}} H_{1}(Np^{r}, \{cusps\}; Z_{p})_{𝔪}^{ord}.

We denote these isomorphic modules by (𝔐_{N})_{𝔪}. As before, there is an action of complex conjugation that yields a decomposition (𝔐_{N})_{𝔪} = (𝔐_{N})_{𝔪}^{+} ⊕ (𝔐_{N})_{𝔪}^{−}.

The following proposition is the Hida-theoretic analogue of Proposition 3.1.1.

Proposition 3.3.1. If ρ̄_{𝔪} is irreducible and p-distinguished, then (𝔐_{N})_{𝔪}^{±} is a free (T_{N})_{𝔪}-module of rank 1.

Proof. Let ℘_{2} denote the product of all classical height one primes ℘ of weight 2 in T_{N} such that κ_{℘} is trivial. Then, by [17, Thm. 3.1],

(𝔐_{N})_{𝔪} ⊗ T_{N}/℘_{2} ≅ (𝔐_{N,1,2})_{𝔪_{2}} (3.4)

where 𝔪_{2} is the maximal ideal in T_{N,1,2} induced by 𝔪. By Proposition 3.1.1 (applied in the case k = 2 and r = 1), we know that (𝔐_{N,1,2})_{𝔪_{2}} is a free T_{N,1,2} ≅ T_{N}/℘_{2}- module of rank 2. Since (𝔐_{N})_{𝔪} ⊗ Q_{p} is free of rank 2 over (T_{N})_{𝔪} ⊗ Q_{p}, it follows that (𝔐_{N})_{𝔪} is free of rank 2 over (T_{N})_{𝔪}. Passing to ±-subspaces then establishes the proposition. □

Using Proposition 3.3.1, we now complete the proof of Proposition 3.1.1 for arbitrary k and r. (Note that the proof of Proposition 3.3.1 only relied upon Proposition 3.1.1 in the already established case of k = 2 and r = 1 and so these arguments are not circular.)

Proof of Proposition 3.1.1. Let ℘_{N,r,k} denote the product of all classical height one primes ℘ of weight k in T_{N} such that κ_{℘} restricted to 1 + p^{r}Z_{p} is trivial. Then, by [19, Theorem 1.9],

(𝔐_{N})_{𝔪} ⊗ T_{N}/℘_{N,r,k} ≅ (𝔐_{N,r,k})_{𝔪_{N,r,k}} (3.5)

where 𝔪_{N,r,k} is the maximal ideal in T_{N,r,k} induced by 𝔪. By Proposition 3.3.1, (𝔐_{N})_{𝔪} is free of rank two over T_{N}. Thus (𝔐_{N,r,k})_{𝔪_{N,r,k}} is free of rank two over T_{N,r,k} ≅ T_{N}/℘_{N,r,k}. Passing to ±-subspace then completes the proof. □

We now assume that the hypotheses of Proposition 3.3.1 are satisfied. Thus we may choose an isomorphism

(3.6)  (T_{N})_{𝔪} ≅^{α_{N}} (𝔐_{N})_{𝔪}^{±}.

Using the notation of the above proof, if we consider α_{N} modulo ℘_{N,r,k}, we get an isomorphism

(T_{N,r,k})_{𝔪_{N,r,k}} ≅^{α_{N,r,k}} (𝔐_{N,r,k}^{±})_{𝔪_{N,r,k}}.

For a given modular form f of level Np^{r} and of weight k in the Hida family corresponding to (T_{N})_{𝔪}, we then define its canonical period to be the one computed with respect to α_{N,r,k}.

Note that the usual ambiguity of the canonical period coming from the choice of isomorphism α_{N,r,k} is controlled across the entire Hida family by the single choice of α_{N} identifying (𝔐_{N})_{𝔪}^{±} with (T_{N})_{𝔪}. In the following section, we show that under these choices of canonical periods, the p-adic L-functions enjoy good congruence properties; that is, they interpolate into a two-variable p-adic L-function.

3.4. Two variable p-adic L-functions. Recall, that for each r ≥ 1, we have a map

P^{1}(Q) → H_{1}(Np^{r}, {cusps}; Z_{p}).

This map is compatible with varying r and so projecting to ordinary parts, localizing at 𝔪 and passing to the limit, we obtain a map

P^{1}(Q) → (𝔐_{N})_{𝔪},

which we again denote by a ↦ {∞, a}.

Definition 3.4.1. For any open subset a + p^{r}Z_{p} of Z_{p}^{×}, we define

μ(a + p^{r}Z_{p}) = U_{p}^{−r}{∞, a/p^{r}} ∈ (𝔐_{N})_{𝔪}.

Proposition 3.4.2. The function μ is a measure (i.e. it is additive).

Proof. This is standard. □

Recall that the completed group ring Z_{p}[[Z_{p}^{×}]] may naturally be regarded as the space of Z_{p}-valued measures on Z_{p}^{×}. We may thus regard μ as defining an element

L(𝔪, N) ∈ (𝔐_{N})_{𝔪} ˆ⊗_{Z_{p}}Z_{p}[[Z_{p}^{×}]].

(Here the tensor product is completed with respect to the usual (i.e. profinite topology) on Z_{p}[[Z_{p}^{×}]] and the 𝔪-adic topology on (𝔐_{N})_{𝔪}.) We may decompose L(𝔪, N) under the action of complex conjugation to get a pair of elements

L^{±}(𝔪, N) ∈ (𝔐_{N})_{𝔪}^{±} ˆ⊗_{Z_{p}}Z_{p}[[Z_{p}^{×}]].

We now assume that the hypotheses of Proposition 3.3.1 are satisfied. Thus we may identify (T_{N})_{𝔪} with (𝔐_{N})_{𝔪}^{±} and so regard

L^{±}(𝔪, N) ∈ (T_{N})_{𝔪} ˆ⊗_{Z_{p}}Z_{p}[[Z_{p}^{×}]] ≅ (T_{N})_{𝔪}[[Z_{p}^{×}]].

We will refer to L^{±}(𝔪, N) as the two-variable p-adic L-function attached to (T_{N})_{𝔪}. Note that it is well-defined up to a unit in (T_{N})_{𝔪}.

If we fix a tame character ω^{i} for some 0 ≤ i ≤ p − 2, then we may project L^{±}(𝔪, N) onto the “ω^{i}-part” of Z_{p}[[Z_{p}^{×}]] and obtain

L(𝔪, N, ω^{i}) ∈ (T_{N})_{𝔪} ˆ⊗_{Z_{p}}Λ_{(i)}.

(Here we choose the sign ± to equal (−1)^{i}, since it is well-known, and easily checked, that otherwise the projection is trivial.)

For any height one prime ℘ in (T_{N})_{𝔪}, we may reduce L^{±}(𝔪, N, ω^{i}) modulo ℘ to obtain an element

L^{±}(𝔪, N, ω^{i})(℘) ∈ ((T_{N})_{𝔪}/℘) ˆ⊗_{Z_{p}}Λ_{(i)} ≅ O(℘) ˆ⊗_{Z_{p}}Λ_{(i)},

which is well-defined up to a unit in O(℘). Also, we may reduce modulo 𝔪 to obtain an element

L̄^{±}(𝔪, N, ω^{i}) ∈ ((T_{N})_{𝔪}/𝔪) ˆ⊗_{Z_{p}}Λ_{(i)} ≅ k ˆ⊗_{Z_{p}}Λ_{(i)},

which is well-defined up to an element of k^{×}.

The following proposition describes the manner in which the two-variable padic L-function L^{±}(𝔪, N, ω^{i}) interpolates the one-variable p-adic L-functions of the modular forms in the Hida family corresponding to (T_{N})_{𝔪}.

Proposition 3.4.3. If ℘ is a classical height one prime ideal in (T_{N})_{𝔪}, then

L^{±}(𝔪, N, ω^{i})(℘) ∈ O(℘) ˆ⊗_{Z_{p}}Λ_{(i)}

is the usual analytic p-adic L-function attached to the corresponding normalized eigenform

f_{℘} ∈ S_{k}(Np^{∞}, O(℘))^{ord}[κ_{℘}]

(computed with respect to the canonical period described in Section 3.3).

Proof. This comparison follows immediately from our given constructions of one and two-variable p-adic L-functions together with the isomorphism in (3.5). □

Remark 3.4.4. Note that the normalized eigenform f_{℘} appearing in the statement of Proposition 3.4.3 need not be a newform and thus L^{±}(𝔪, N, ω^{i})(℘) may not be a “primitive” p-adic L-function. This issue will be explored in the following section.

Remark 3.4.5. The following remark answers a question asked by the referee; it will not be used elsewhere in the paper.

Note that the formula of Definition 3.4.1 works equally well to define an (𝔐_{N})_{𝔪}- valued measure on Z_{p}. Integrating the constant function 1 on Z_{p} with respect to this measure yields an element of (𝔐_{N})_{𝔪}; explicitly, this is just the element {∞, 1} ∈ (𝔐_{N})_{𝔪}. This element in fact lies in (𝔐_{N})_{𝔪}^{+}, and thus (via our chosen identification of (𝔐_{N})_{𝔪}^{+} with (T_{N})_{𝔪}) gives rise to an element of (T_{N})_{𝔪}, which we denote by L^{∗}(𝔪, N, 1). Any easy computation (using the fact that Z_{p} = Z_{p}^{×} ⋃ pZ_{p}) shows that (1 − U_{p}^{−1})L^{∗}(𝔪, N, 1) = L^{+}(𝔪, N, 1). (Note that the ‘1’ in the third argument stands for the trivial character ω^{0}.) Thus L^{∗}(𝔪, N, 1) coincides with the “improved p-adic L-function” of [14], computed with respect to a canonical period.

3.5. Two variable p-adic L-functions on branches of the Hida family. Fix an irreducible modular Galois representation ρ̄ : G_{Q} → GL_{2}(k) that is both pordinary and p-distinguished. We choose also a p-stabilization on ρ̄ (which we will suppress in our discussion and notation). Furthermore, fix a finite set of primes Σ and consider the Hecke algebra T_{Σ} associated to ρ̄ as in Section 2.4.

Proposition 2.4.2 yields an isomorphism T_{Σ} ≅ (T_{N(Σ)})_{𝔫} for a certain maximal ideal 𝔫 of T_{N(Σ)}. The construction of the preceding section thus defines a twovariable p-adic L-function

L(𝔫, N(Σ), ω^{i}) ∈ (T_{N(Σ)})_{𝔫} ˆ⊗_{Z_{p}}Λ_{(i)}.

The isomorphism T_{Σ} ≅ (T_{N(Σ)})_{𝔫} then yields elements

L_{Σ}(ρ̄, ω^{i}) ∈ T_{Σ} ˆ⊗_{Z_{p}}Λ_{(i)},
L_{Σ}(ρ̄, ω^{i})(℘) ∈ O(℘) ⊗_{Z_{p}} Λ_{(i)},
\overline{L}_{Σ}(ρ̄, ω^{i}) ∈ k ˆ⊗_{Z_{p}}Λ_{(i)},

as defined in the previous section, which are well-defined up to units in T_{Σ}, O(℘) and k respectively.

We now note that there are two different ways that one can naturally attach a modular form to a classical height one prime ideal ℘ of T_{Σ}. First, via the isomorphism of Proposition 2.4.2, this prime corresponds to a classical height one prime ideal of T_{N(Σ)} which we again call ℘. By part (2) of Theorem 2.1.1, this prime gives rise to a normalized eigenform f_{℘} of tame level N(Σ). On the other hand, by Proposition 2.5.6, ℘ also corresponds to a classical height one prime of T_{N(𝔞)}^{new} which we denote by ℘^{new}. (Here 𝔞 is the unique minimal prime of T_{Σ} contained in ℘.) By part (2) of Theorem 2.1.3, this prime gives rise to a normalized newform f_{℘}^{new} := f_{℘^{new}} of tame level N(𝔞).

In general, f_{℘} and f_{℘}^{new} are not equal. Indeed, they are eigenforms for different Hecke algebras: f_{℘}^{new} is an eigenform for T_{N(𝔞)} while f_{℘} is an eigenform for T_{N(Σ)}. Proposition 2.4.2 makes it clear how they are related: f_{℘} is the normalized oldform obtained from f_{℘}^{new} by “removing the Euler factor” at each of the primes ℓ ∈ Σ.

Note that Proposition 3.4.3 shows that L_{Σ}(ρ̄, ω^{i}) interpolates the p-adic Lfunctions attached to the modular forms f_{℘} for ℘ a classical height one prime of T_{Σ}. In particular, these are “non-primitive” p-adic L-functions.

We now seek to construct a two variable p-adic L-function for each minimal prime 𝔞 of T_{Σ}, which interpolates the p-adic L-functions of the modular forms f_{℘}^{new} as ℘^{new} runs through the classical height one primes of T(𝔞). For this, we recall the natural isomorphism T(𝔞)^{∘} ≅ T_{N(𝔞)}^{new}/𝔞′ of Definition 2.5.3, which gives rise to the composite surjection

(3.7)  T_{N(𝔞)} → T_{N(𝔞)}^{new} → T_{N(𝔞)}^{new}/𝔞′ ≅ T(𝔞)^{∘}.

If we let 𝔪 denote the maximal ideal of T_{N(𝔞)} obtained as the preimage of the maximal ideal of T(𝔞)^{∘} under (3.7), then the construction of the preceding section yields an L-function L(𝔪, N(𝔞), ω^{i}) ∈ T_{N(𝔞)} ˆ⊗_{Z_{p}}Λ_{(i)}. The surjection (3.7) induces a corresponding surjection

T_{N(𝔞)} ˆ⊗_{Z_{p}}Λ_{(i)} → T(𝔞)^{∘} ˆ⊗_{Z_{p}}Λ_{(i)}.

We then denote the image of L(𝔪, N(𝔞), ω^{i}) under this surjection by

L(ρ̄, 𝔞, ω^{i}) ∈ T(𝔞)^{∘} ˆ⊗_{Z_{p}}Λ_{(i)}

and we refer to L(ρ̄, 𝔞, ω^{i}) as the two variable p-adic L-function attached to T(𝔞). The following proposition describes the interpolation property of L(ρ̄, 𝔞, ω^{i}).

Proposition 3.5.1. If ℘^{new} is a classical height one prime ideal in T(𝔞),

L(ρ̄, 𝔞, ω^{i})(℘^{new}) ∈ O(℘^{new}) ˆ⊗_{Z_{p}}Λ_{(i)}

equals the p-adic L-function of f_{℘}^{new} := f_{℘^{new}} computed with respect to a canonical period.

Proof. This proposition follows immediately from Proposition 3.4.3. □

3.6. Comparisons. We now have a two-variable p-adic L-function L_{Σ}(ρ̄, ω^{i}) defined over the entire Hida algebra T_{Σ} and, for each minimal prime 𝔞 of T_{Σ}, we have a two-variable p-adic L-function L(ρ̄, 𝔞, ω^{i}) defined on the branch T(𝔞). It is natural then to compare L(ρ̄, 𝔞, ω^{i}) with L_{Σ}(ρ̄, ω^{i}) modulo 𝔞.

Each of these p-adic L-functions lies in T(𝔞)^{∘} ˆ⊗_{Z_{p}}Λ_{(i)}. The interpolation property satisfied by classical p-adic L-functions shows that for any height one prime ℘ of T_{Σ} containing 𝔞, the L-functions L_{p}(f_{℘}^{new}) and L_{p}(f_{℘}) agree up to multiplication by a certain product of Euler factors (reflecting the Euler factors removed from f_{℘}^{new} to obtain f_{℘}) and an element of O(℘)[1/p]^{×} (reflecting the ratio of the canonical period of f_{℘}^{new} and the canonical period of f_{℘}).

We will show that in fact the ratio of these canonical periods lies in (O(℘)^{∘})^{×}. Hence, L_{p}(f_{℘}^{new}) and L_{p}(f_{℘}) agree up to multiplication by a product of Euler factors and the inevitable ambiguity of a unit of O(℘)^{∘}. Furthermore, we will show that this occurs uniformly in Hida families; in other words, that L_{Σ}(ρ̄, ω^{i}) mod 𝔞 and L(ρ̄, 𝔞, ω^{i}) agree up to multiplication by a product of Euler factors and a unit in T(𝔞)^{∘}.

We begin by defining the relevant Euler factors. Recall the Euler factors E_{ℓ}(𝔞, X) in T(𝔞)^{∘}[X] defined in Section 2.7. Since ℓ is prime to p, we may regard ℓ as an element of Z_{p}^{×} and so obtain a corresponding unit element ⟨ℓ⟩_{p} ∈ Z_{p}[[Z_{p}^{×}]]. Thus

E_{ℓ}(𝔞, ⟨ℓ⟩_{p}^{−1}) ∈ T(𝔞)^{∘}[[Z_{p}^{×}]] ≅ T(𝔞)^{∘} ˆ⊗_{Z_{p}}Z_{p}[[Z_{p}^{×}]].

For 0 ≤ i ≤ p − 2, we write ⟨ℓ⟩_{p,i} to denote the projection of ⟨ℓ⟩_{p} ∈ Z_{p}[[Z_{p}^{×}]] under the surjection Z_{p}[[Z_{p}^{×}]] → Λ_{(i)}. We write E_{ℓ}(𝔞, ⟨ℓ⟩_{p,i}^{−1}) to denote the corresponding element of T(𝔞)^{∘} ˆ⊗_{Z_{p}}Λ_{(i)}.

Definition 3.6.1. If 𝔞 is a minimal prime of T_{Σ}, then we write

E_{Σ}(𝔞) := ∏_{ℓ∈Σ} E_{ℓ}(𝔞, ⟨ℓ⟩_{p}^{−1}) ∈ T (𝔞)^{∘} ˆ⊗_{Z_{p}}Z_{p}[[Z_{p}^{×}]]

and

E_{Σ}(𝔞, ω^{i}) := ∏_{ℓ∈Σ} E_{ℓ}(𝔞, ⟨ℓ⟩_{p,i}^{−1}) ∈ T (𝔞)^{∘} ˆ⊗_{Z_{p}}Λ_{(i)}.

Since L_{Σ}(ρ̄, ω^{i}) and L(ρ̄, 𝔞, ω^{i}) are constructed using modular symbols of levels N(Σ) and N(𝔞) respectively, in order to compare them, we will need to be able to compare the corresponding Hecke algebras. Recall that by construction T_{Σ} = (T′_{N(Σ)})_{𝔪′} for a certain maximal ideal 𝔪′ of T′_{N(Σ)} and that 𝔫 is a certain maximal ideal of T_{N(Σ)} lying over 𝔪′ with the property that the map (T′_{N(Σ)})_{𝔪′} → (T_{N(Σ)})_{𝔫} is an isomorphism. We let 𝔪 denote the preimage in T_{N(𝔞)}, under the surjection (3.7), of the maximal ideal of T(𝔞)^{∘}. Altogether, we have the following diagram of maps between the various Hecke algebras:

(3.8)  \xymatrix{(T′_{N(Σ)})_{𝔪′} \ar[d] \ar[r]^{≅} & (T_{N(Σ)})_{𝔫} \\ (T_{N(𝔞)})_{𝔪} \ar[r] & T(𝔞)^{∘}.}

By inverting the upper horizontal isomorphism, we obtain a map (T_{N(Σ)})_{𝔫} → T(𝔞)^{∘}. This is a homomorphism of T′_{N(Σ)}-algebras when we equip the source and target with the T′_{N(Σ)}-algebra structure provided by the diagram (3.8).

We write (𝔐_{N(Σ)})_{𝔫} and (𝔐_{N(𝔞)})_{𝔪} to denote the modules of p-adic modular symbols constructed in Section 3.3 for the indicated choice of tame level and localized at the indicated maximal ideal. The following result provides the key to comparing the L-functions L_{Σ}(ρ̄, ω^{i}) mod 𝔞 and L(ρ̄, 𝔞, ω^{i}). We postpone its proof to the Section 3.8.

Theorem 3.6.2. There is an isomorphism of T(𝔞)^{∘}-modules

T(𝔞)^{∘} ⊗_{(T_{N(Σ)})_{𝔫}} (𝔐_{N(Σ)})_{𝔫} ≅ T(𝔞)^{∘} ⊗_{(T_{N(𝔞)})_{𝔪}} (𝔐_{N(𝔞)})_{𝔪},

compatible with the action of complex conjugation.

Moreover, the induced map on the corresponding spaces of measures taking values in these modules sends the measure L(𝔫, N(Σ)), which takes values in the source of the isomorphism, to the measure L(𝔪, N(𝔞))E_{Σ}(𝔞), which takes values in the target of the isomorphism.

Corollary 3.6.3. There is a unit u ∈ T(𝔞)^{∘} such that

L_{Σ}(ρ̄, ω^{i}) ≡ u · L(ρ̄, 𝔞, ω^{i})E_{Σ}(𝔞, ω^{i}) (mod 𝔞).

Proof. This follows immediately from the preceding theorem. □

3.7. Iwasawa invariants. If we choose an isomorphism

(3.9)  Λ ≅ Z_{p}[[T]],

and hence an isomorphism (T_{N})_{𝔪} ˆ⊗_{Z_{p}}Λ_{(i)} ≅ (T_{N})_{𝔪}[[T]], we may then regard L(𝔪, N, ω^{i}) as an element of T_{𝔪}[[T]]. We are, of course, interested in the Iwasawa invariants of such power series. We begin this section by reviewing the definitions and basic properties of such invariants.

Definition 3.7.1. If R is a ring and f(T) ∈ R[[T]] is a one-variable power series with coefficients in R, then we define the content of f(T) to be the ideal I(f(T)) ⊂ R generated by the coefficients of f(T).

The proof of the next lemma is straightforward.

Lemma 3.7.2. Let f(T) ∈ R[[T]].

(1) If α is any automorphism of R[[T]], then I(f(T)) = I(α(f(T))).

(2) If u(T) ∈ R[[T]]^{×}, then I(u(T)f(T)) = I(f(T)).

(3) If f(T) = g(T)h(T) in R[[T]], then if any two of these three power series have unit content, so does the third.

(4) If ϕ : R → S is a morphism of rings then I(ϕ(f(T))) = ϕ(I(f(T))).

In particular, regarding L(𝔪, N, ω^{i}) as a power series, we may consider its content I(L(𝔪, N, ω^{i})), an ideal of (T_{N})_{𝔪}. Parts (1) and (2) of the preceding lemma show that this ideal is in fact independent of the choice of the isomorphisms (3.6) and (3.9).

For each height one prime ℘, we may likewise form the corresponding ideal I(L(𝔪, N, ω^{i})(℘)) in O(℘). By part (4) of Lemma 3.7.2, this ideal equals the image of I(L(𝔪, N, ω^{i})) under the surjection (T_{N})_{𝔪} → (T_{N})_{𝔪}/℘ ≅ O(℘). We also have that I(L̄(𝔪, N, ω^{i})) is an ideal of k and is thus either zero or all of k. Again, we have that this ideal equals the image of I(L(𝔪, N, ω^{i})) under the surjection (T_{N})_{𝔪} → (T_{N})_{𝔪}/𝔪 ≅ k.

The next result is an immediate consequence of the fact that (T_{N})_{𝔪} is a local ring.

Proposition 3.7.3. Fix i such that 0 ≤ i ≤ p − 2. The following are equivalent: (1) I(L(𝔪, N, ω^{i})) = (T_{N})_{𝔪}.

(2) I(L(𝔪, N, ω^{i})(℘)) = O(℘) for some height one prime ℘ of (T_{N})_{𝔪}.

(3) I(L(𝔪, N, ω^{i})(℘)) = O(℘) for every height one prime ℘ of (T_{N})_{𝔪}.

(4) I(L̄(𝔪, N, ω^{i})) is non-zero (and hence equal to k).

We remark that if ℘ is a classical height one prime ideal in (T_{N})_{𝔪}, then the length of the quotient O(℘)/I(L(𝔪, N, ω^{i})(℘)) is related to the usual μ-invariant of the p-adic L-function L(𝔪, N, ω^{i})(℘) ∈ O(℘)[[T]]. In particular, this μ-invariant vanishes if and only if I(L(𝔪, N, ω^{i})(℘)) is the unit ideal of O(℘).

Analogously, we have well-defined ideals

I(L_{Σ}(ρ̄, ω^{i})), I(L_{Σ}(ρ̄, ω^{i})(℘)) and I(\overline{L}_{Σ}(ρ̄, ω^{i}))

of T_{Σ}, O(℘) and k respectively. The above result applies equally well to these ideals since they are constructed out of I(L(𝔪, N, ω^{i})) for some choice of 𝔪 and N.

We now compute the content of the reciprocal Euler factor E_{Σ}(𝔞, ω^{i}) defined in Section 3.6.

Lemma 3.7.4. The element E_{Σ}(𝔞, ω^{i}) of T (𝔞)^{∘} ˆ⊗_{Z_{p}}Λ_{(i)} has unit content.

Proof. Part (3) of Lemma 3.7.2 shows that it suffices to prove that each of the reciprocal Euler factors E_{ℓ}(𝔞, ⟨ℓ⟩_{p,i}^{−1}) has unit content. If γ is a topological generator of Γ, then we may write ℓ^{−1}ω(ℓ) = γ^{up^{n}} for some u ∈ Z_{p}^{×} and some integer n ≥ 0. If we choose our isomorphism (3.9) so that ⟨γ⟩_{p} ↦ 1 + T, then we see that

E_{ℓ}(𝔞, ⟨ℓ⟩_{p,i}^{−1}) =

(1 − (T_{ℓ} mod 𝔞′)ω^{−i}(ℓ)(1 + T)^{up^{n}} + ⟨ℓ⟩_{N(𝔞)p}ω^{−2i}(ℓ)ℓ^{−1}(1 + T)^{2up^{n}})

if ℓ is prime to N(𝔞) (1 − (T_{ℓ} mod 𝔞′)ω^{−i}(ℓ)(1 + T)^{up^{n}}) otherwise.

In the second case either the constant term is a unit (if T_{ℓ} is not a unit) or else the coefficient of T^{p^{n}} is a unit (if T_{ℓ} is a unit). In the first case, we may compute the content after making the substitution (1 + T)^{u} ↦ 1 + T, in which case one sees immediately that the coefficient of T^{2p^{n}} is a unit. □

The following theorem establishes that if the μ-invariant vanishes for one form in a Hida family, then it vanishes for every form in that family.

Theorem 3.7.5. The following are equivalent:

(1) μ(L_{p}(f, ω^{i})) = 0 for some newform f in the Hida family of ρ̄.

(2) μ(L_{p}(f, ω^{i})) = 0 for every newform f in the Hida family of ρ̄.

(3) L(ρ̄, 𝔞, ω^{i}) has unit content for some irreducible component T(𝔞) of the Hida family of ρ̄.

(4) L(ρ̄, 𝔞, ω^{i}) has unit content for every irreducible component T(𝔞) of the Hida family of ρ̄.

Proof. Let T(𝔞) be an irreducible component of the Hida family of ρ̄ and let ℘ be a classical height one prime ideal of T(𝔞). Since L(ρ̄, 𝔞, ω^{i}) modulo ℘ equals L_{p}(f_{℘}, ω^{i}) and T(𝔞)^{∘} is a local ring, we deduce from Proposition 3.7.3 that the following are equivalent: L(ρ̄, 𝔞, ω^{i}) has unit content; L_{p}(f_{℘}, ω^{i}) has unit content (i.e. trivial μ-invariant) for one classical height one prime ℘ of T(𝔞); L_{p}(f_{℘}, ω^{i}) has unit content for every classical height one prime ℘ of T(𝔞).

To complete the proof of the proposition, it suffices to show that if T(𝔞_{1}) and T(𝔞_{2}) are two irreducible components of the Hida family of ρ̄, then L(ρ̄, 𝔞_{1}, ω^{i}) has unit content if and only if the same holds true for L(ρ̄, 𝔞_{2}, ω^{i}). We may choose Σ large enough so that each of T(𝔞_{1}) and T(𝔞_{2}) is an irreducible component of T_{Σ}. Corollary 3.6.3, together with Lemmas 3.7.2 (3) and 3.7.4, then shows that each of L(ρ̄, 𝔞_{1}, ω^{i}) and L(ρ̄, 𝔞_{2}, ω^{i}) have unit content if and only if the same is true of L_{Σ}(ρ̄, ω^{i}). □

If the equivalent conditions of the preceding theorem hold, then we write

μ^{an}(ρ̄, ω^{i}) = 0.

(As usual, our notation suppresses the choice of p-stabilization of ρ̄.) In the case when the μ-invariant vanishes we can further study the λ-invariant of these power series. (Note that for a local ring which is not a discrete valuation ring one can only define the λ-invariant for power series of unit content.)

Definition 3.7.6. If A is a local ring and f(T) ∈ A[[T]] is a power series having unit content, then we define the λ-invariant λ(f(T)) to be the smallest degree in which f(T) has a unit coefficient.

We remark that if ϕ : A → B is a local morphism of complete local rings and if f(T) is an element of A[[T]] having unit content, then λ(f(T)) = λ(ϕ(f(T))).

If T(𝔞) is an irreducible component of the Hida family of ρ̄ with μ^{an}(ρ̄, ω^{i}) = 0, then Theorem 3.7.5 shows that L(ρ̄, 𝔞, ω^{i}) has unit content. We can therefore define the analytic λ-invariant of a branch by

λ^{an}(ρ̄, 𝔞, ω^{i}) = λ(L(ρ̄, 𝔞, ω^{i})).

Our main results on analytic λ-invariants in Hida families are as follows.

Theorem 3.7.7. Assume that μ^{an}(ρ̄, ω^{i}) = 0.

(1) For any given irreducible component T(𝔞) of the Hida family of ρ̄, the λinvariant of L_{p}(f_{℘}, ω^{i}) takes on the constant value of λ^{an}(𝔞, ω^{i}) as ℘ varies over all classical height one primes of T(𝔞).

(2) For any two irreducible components T(𝔞_{1}), T(𝔞_{2}) of the Hida family of ρ̄, we have that

λ^{an}(𝔞_{1}, ω^{i}) − λ^{an}(𝔞_{2}, ω^{i}) = ∑_{ℓ≠p}(e_{ℓ}(𝔞_{2}, ω^{i}) − e_{ℓ}(𝔞_{1}, ω^{i}))

where e_{ℓ}(𝔞, ω^{i}) = λ(E_{ℓ}(𝔞, ⟨ℓ⟩_{p,i}^{−1})).

Proof. The first part follows from Proposition 3.5.1 and Definition 3.7.6. For the second part, choose Σ large enough so that both T(𝔞_{1}) and T(𝔞_{2}) are irreducible components of T_{Σ}. Then, by Corollary 3.6.3, we have that

λ(L_{Σ}(ρ̄, ω^{i})) = λ^{an}(𝔞_{j}, ω_{i}) + ∑_{ℓ∈Σ} e_{ℓ}(𝔞_{j}, ω^{i})

for j = 1, 2. Our formula then follows since e_{ℓ}(𝔞_{1}, ω^{i}) = e_{ℓ}(𝔞_{2}, ω^{i}) for ℓ ∉ Σ. □

3.8. Proof of Theorem 3.6.2. In this section, we present the proof of Theorem 3.6.2. We will utilize the notation introduced prior to the statement of the theorem and begin the proof by introducing some additional notation.

If M ≥ 1 is an integer, d is a divisor of M and d′ is a divisor of d, then we let B_{d,d′} : X_{1}(M) → X_{1}(M/d) denote the map induced by the map τ ↦ d′τ on the upper half-plane. This map induces a corresponding map

(B_{d,d′})_{∗} : H_{1}(M, {cusps}; Z_{p}) → H_{1}(M/d, {cusps}; Z_{p})

on relative homology groups.

If ℓ is any prime distinct from p, then we let e_{ℓ} denote the largest power of ℓ dividing N(Σ)/N(𝔞). We have the inequality 0 ≤ e_{ℓ} ≤ 2. Also, e_{ℓ} = 0 unless ℓ ∈ Σ, while e_{ℓ} = 2 only if ℓ ∈ Σ and ℓ is prime to N(𝔞). For each such prime ℓ, we write

ϵ(ℓ) := \begin{cases}1 & if e_{ℓ} = 0 \\ ((B_{ℓ,1})_{∗} − ℓ^{−1}T_{ℓ} (B_{ℓ,ℓ})_{∗}) & if e_{ℓ} = 1 \\ ((B_{ℓ^{2},1})_{∗} − ℓ^{−1}T_{ℓ} (B_{ℓ^{2},ℓ})_{∗} + ℓ^{−3}⟨ℓ⟩_{N(𝔞)p}(B_{ℓ^{2},ℓ^{2}})_{∗}) & if e_{ℓ} = 2. \end{cases}

Now write Σ = {ℓ_{1}, …, ℓ_{n}} and for any r ≥ 1 define

ϵ_{r} : H_{1}(N(Σ)p^{r}, {cusps}; Z_{p})^{ord} → H_{1}(N(𝔞)p^{r}, {cusps}; Z_{p})^{ord}

by ϵ_{r} = ϵ(ℓ_{n}) ∘ ⋯ ∘ ϵ(ℓ_{1}).

Let us explain this formula. For any i = 1, …, n, write N_{i} = N(Σ)/ℓ_{1}^{e_{ℓ_{1}}} ⋯ ℓ_{i}^{e_{ℓ_{i}}} . In the formula for ϵ_{r}, the map ϵ(ℓ_{i}) is taken to be the map

H_{1}(N_{i−1}p^{r}, {cusps}; Z_{p})^{ord} → H_{1}(N_{i}p^{r}, {cusps}; Z_{p})^{ord}

given by the stated formula for ϵ(ℓ_{i}). (The symbol T_{ℓ_{i}} in the formula for ϵ(ℓ_{i}) is understood to stand for the corresponding Hecke operator acting in level N_{i}p^{r}.) It is easily verified that the map ϵ_{r} is in fact independent of the choice of ordering of the elements of Σ.

For any tame level M we let (T_{M}^{∗})′ denote the Λ-subalgebra of the ordinary Hecke algebra T_{M}^{∗} generated by the Hecke operators prime to M. If we regard the source and target of ϵ_{r} as (T_{N(Σ)}^{∗})′-modules via the inclusion (T_{N(Σ)}^{∗})′ ⊂ T_{N(Σ)}^{∗} and the natural map (T_{N(Σ)}^{∗})′ → (T_{N(𝔞)}^{∗})′ ⊂ T_{N(𝔞)}^{∗}, then ϵ_{r} is immediately seen to be (T_{N(Σ)}^{∗})′-linear.

As r varies, the sources and targets of the maps ϵ_{r} each form a projective system and the maps ϵ_{r} are evidently compatible with the projection maps on source and target. Thus, passing to the limit in r, we obtain a (T_{N(Σ)}^{∗})′-linear map

ϵ_{∞} : lim_{\overleftarrow{r}} H_{1}(N(Σ)p^{r}, \{cusps\}; Z_{p})^{ord} → lim_{\overleftarrow{r}} H_{1}(N(𝔞)p^{r}, \{cusps\}; Z_{p})^{ord}.

We denote the source and target of this map by 𝔐_{N(Σ)} and 𝔐_{N(𝔞)} respectively.

We may regard each of the maximal ideals 𝔪′, 𝔫 and 𝔪 equally well as maximal ideals of (T_{N(Σ)}^{∗})′, T_{N(Σ)}^{∗} and T_{N(𝔞)}^{∗}. If we localize ϵ_{∞} with respect to 𝔪′, we obtain a map

(3.10)  (𝔐_{N(Σ)})_{𝔪′} → (𝔐_{N(𝔞)})_{𝔪′}.

Now 𝔫 and 𝔪 each pull back to 𝔪′ under the natural maps T′_{N(Σ)} → T_{N(Σ)} and T′_{N(Σ)} → T_{N(𝔞)} and so the localizations (T_{N(Σ)})_{𝔫} and (T_{N(𝔞)})_{𝔪} are local factors of the complete semi-local rings (T′_{N(Σ)})_{𝔪′} ⊗_{T′_{N(Σ)}} T_{N(Σ)} and (T′_{N(Σ)})_{𝔪′} ⊗_{T′_{N(Σ)}} T_{N(𝔞)}. Thus the localizations (𝔐_{N(Σ)})_{𝔫} and (𝔐_{N(𝔞)})_{𝔪} appear naturally as direct factors of (𝔐_{N(Σ)})_{𝔪′} and (𝔐_{N(𝔞)})_{𝔪′} respectively, and so the map (3.10) induces a map

(𝔐_{N(Σ)})_{𝔫} → (𝔐_{N(𝔞)})_{𝔪}.

Tensoring the source of this map with T(𝔞)^{∘} over (T_{N(Σ)})_{𝔫} and the target with T(𝔞)^{∘} over (T_{N(𝔞)})_{𝔪}, we obtain a T(𝔞)^{∘}-linear map

(3.11)  T(𝔞)^{∘} ⊗_{(T_{N(Σ)})_{𝔫}} (𝔐_{N(Σ)})_{𝔫} → T(𝔞)^{∘} ⊗_{(T_{N(𝔞)})_{𝔪}} (𝔐_{N(𝔞)})_{𝔪}.

We claim that this map satisfies the requirements of Theorem 3.6.2.

We first observe that this map satisfies the claimed property with regard to the L-functions. This follows from an explicit calculation of the effect of the maps ϵ_{ℓ} on modular symbols. In fact, one easily shows that for any character χ of conductor

p^{n} and α = ∑_{a∈(Z/p^{n})^{×}} χ(a) {a/p^{n}, ∞}, we have that

ϵ_{r} (α) =

∏_{i s.t. e_{ℓ_{i}} =1}(1 − χ^{−1}(ℓ)ℓ^{−1}T_{ℓ}) ∏_{i s.t. e_{ℓ_{i}} =2}(1 − χ^{−1}(ℓ)ℓ^{−1}T_{ℓ} + χ^{−2}(ℓ)ℓ^{−3}⟨ℓ⟩_{N(𝔞)p}) · α.

(In the left hand side of this equation, the modular symbols {a/p^{n}, ∞} are regarded as lying in H_{1}(N(Σ)p^{r}, {cusps}; Z_{p}). On the right hand side, they are regarded as lying in H_{1}(N(𝔞)p^{r}, {cusps}; Z_{p}).) Passing to the limit in r, and taking into account the fact that χ is an arbitrary Dirichlet character of p-power conductor, we conclude the the isomorphism (3.11) has the required effect on L-functions.

We now turn to showing that (3.11) is an isomorphism. Note that Proposition 3.3.1 shows that both source and target are free of rank two over T(𝔞)^{∘}. Thus to see that this map is an isomorphism, it suffices to check that it induces a surjection after being reduced modulo the maximal ideal of T(𝔞)^{∘}. To do this, we first mod out by a classical prime of weight two and then by the full maximal ideal.

Let ℘ denote the classical height one prime in Λ of weight two for which κ_{℘} is trivial. If we tensor each side of (3.11) by Λ/℘ over Λ, we obtain the map

(3.12) T(𝔞)^{∘} ⊗_{(T_{N(Σ)})_{𝔫}} H_{1}(N(Σ)p; Z_{p})_{𝔫}^{ord} → T(𝔞)^{∘} ⊗_{(T_{N(𝔞)})_{𝔪}} H_{1}(N(𝔞)p; Z_{p})_{𝔪}^{ord}

induced by localizing the source and target of ϵ_{1} at 𝔫 and 𝔪 respectively and then extending scalars to T(𝔞)^{∘}. (Here, as in Proposition 3.3.1, we are using [17, Thm. 3.1].)

Thus the reduction modulo the maximal ideal of T(𝔞)^{∘} of (3.11) coincides with the map

(3.13) (T_{N(𝔞)}/𝔪) ⊗_{T_{N(Σ)}/𝔫} (H_{1}(N(Σ)p; Z_{p})^{ord}/𝔫H_{1}(N(Σ)p; Z_{p})^{ord})

→ H_{1}(N(𝔞)p; Z_{p})^{ord}/𝔪H_{1}(N(𝔞)p; Z_{p})^{ord}

of T_{N(𝔞)}/𝔪-vector spaces induced by (3.12).

Rather than showing directly that (3.13) is surjective, we will show that the corresponding dual map

(3.14) H^{1}(N(𝔞)p; F_{p})^{ord}[𝔪] → (T_{N(𝔞)}/𝔪) ⊗_{T_{N(Σ)}/𝔫} (H^{1}(N(Σ)p; F_{p})^{ord}[𝔫])

is injective. (In writing the dual of (3.13) in this form, we have implicitly fixed an isomorphism of one dimensional T_{N(𝔞)}/𝔪-vector spaces between T_{N(𝔞)}/𝔪 and its T_{N(Σ)}/𝔪′-linear dual. We will suppress this choice of isomorphism here and below.) This map may be written as a composite

(3.15)

H^{1}(N(𝔞)p; F_{p})^{ord}[𝔪] → (T_{N(𝔞)}/𝔪) ⊗_{T_{N(Σ)}/𝔫} (H^{1}(N(𝔞)p; F_{p})^{ord}[𝔪′])
→ (T_{N(𝔞)}/𝔪) ⊗_{T_{N(Σ)}/𝔫} (H^{1}(N(Σ)p; F_{p})^{ord}[𝔪′])
→ (T_{N(𝔞)}/𝔪) ⊗_{T_{N(Σ)}/𝔫} (H^{1}(N(Σ)p; F_{p})^{ord}[𝔫]) .

To explain this, we first recall that since the localization (T_{N(Σ)})_{𝔫} is a local factor of the tensor product (T′_{N(Σ)})_{𝔪′} ⊗_{T′_{N(Σ)}} T_{N(Σ)} for which the natural map (T′_{N(Σ)})_{𝔪′} → (T_{N(Σ)})_{𝔫} is an isomorphism, the residue field T_{N(Σ)}/𝔫 is a local factor of the Artin local ring T_{N(Σ)}/𝔪′T_{N(Σ)}; write

T_{N(Σ)}/𝔪′T_{N(Σ)} ≅ T_{N(Σ)}/𝔫 × A,

where A denotes the product of the remaining local factors. This decomposition induces a corresponding decomposition

(3.16)  B ≅ B[𝔫] × A ⊗_{T_{N(Σ)}/𝔪′T_{N(Σ)}} B

for any T_{N(Σ)}/𝔪′T_{N(Σ)}-module B. The third arrow of (3.15) is precisely projection onto the the first of the two factors in (3.16) with

B = (T_{N(𝔞)}/𝔪) ⊗_{T_{N(Σ)}/𝔫} (H^{1}(N(Σ)p; F_{p})^{ord}[𝔪′]) .

The second arrow is induced by dualizing the reduction modulo 𝔪′ of the map (3.10) and the first arrow is induced from the obvious inclusion, or if the reader prefers, is obtained by dualizing the surjection

(T_{N(𝔞)}/𝔪) ⊗_{T_{N(Σ)}/𝔫} (H_{1}(N(𝔞)p; Z_{p})^{ord}/𝔪′H_{1}(N(𝔞)p; Z_{p})^{ord})

→ H_{1}(N(𝔞)p; Z_{p})^{ord}/𝔪H_{1}(N(𝔞)p; Z_{p})^{ord}.

Lemma 3.8.1. The map (3.14) is injective if and only if the composite of the first two arrows of (3.15) is injective.

Proof. The only if statement is clear. In order to prove the other statement, we first observe that the second arrow of (3.15) is given by the cohomological version of the map ϵ_{1}. More precisely, if for each ℓ ∈ Σ we define

ϵ_{ℓ}^{∗} := \begin{cases}1 & if e_{ℓ} = 0 \\ B_{ℓ,1}^{∗} − B_{ℓ,ℓ}^{∗} ℓ^{−1}T_{ℓ} & if e_{ℓ} = 1 \\ B_{ℓ^{2},1}^{∗} − B_{ℓ^{2},ℓ}^{∗} ℓ^{−1}T_{ℓ} + B_{ℓ^{2},ℓ^{2}}^{∗}ℓ^{−3}⟨ℓ⟩_{N(𝔞)p} & if e_{ℓ} = 2 \end{cases}

(where B_{d,d′}^{∗} denotes the map on cohomology induced by the degeneracy map B_{d,d′}) and define

ϵ^{∗} = ϵ_{ℓ_{1}}^{∗} ∘ ⋯ ∘ ϵ_{ℓ_{n}}^{∗} : H^{1}(N(𝔞)p; F_{p})^{ord} → H^{1}(N(Σ)p; F_{p})^{ord},

then the second arrow of (3.15) is obtained from the map ϵ^{∗} by passing to the kernel of 𝔪′ in the source and target and then extending scalars from T_{N(Σ)}/𝔫 to T_{N(𝔞)}/𝔪.

One then checks that any element in the image of ϵ^{∗} is annihilated by the Hecke operators T_{ℓ}, for those ℓ ∈ Σ. On the other hand, Proposition 2.4.2 shows that the maximal ideal 𝔫 is uniquely characterized by the property of containing these operators. Thus the image of the second arrow of (3.15) lies in the local factor

(T_{N(𝔞)}/𝔪) ⊗_{T_{N(Σ)}/𝔫} (H_{1}(N(Σ)p; F_{p})^{ord}[𝔫])

of

(T_{N(𝔞)}/𝔪) ⊗_{T_{N(Σ)}/𝔫} (H_{1}(N(Σ)p; F_{p})^{ord}[𝔪′]) .

This proves the lemma. □

By the preceding lemma, we are reduced to proving that the composite of the first two arrows of (3.15) is injective. It will be notationally easier to deal with each of the maps ϵ_{ℓ_{i}} separately and so we put ourselves in the following more general situation. We consider a natural number M prime to p and a prime ℓ distinct from p. We define n_{ℓ} = 1 or 2 according to whether or not ℓ divides M and write N := ℓ^{n_{ℓ}}M. We let 𝔪 denote a maximal ideal of T_{M} for which ρ̄_{𝔪} is irreducible and p-distinguished and let 𝔪′ denote the pull back of 𝔪 under the natural map T′_{N} → T_{M}. The map ϵ_{ℓ}^{∗} defined in the proof of Lemma 3.8.1 induces a T_{M}/𝔪-linear map

(3.17)  H^{1}(Mp; F_{p})^{ord}[𝔪] → (T_{M}/𝔪) ⊗_{T′_{N} /𝔪′} (H^{1}(Np; F_{p})^{ord}[𝔪′]) .

Lemma 3.8.2. The map (3.17) is injective.

The proof of Lemma 3.8.1 shows that elements in the image of (3.17) are annihilated by T_{ℓ} and hence are in fact annihilated by a maximal ideal of the full Hecke algebra T_{N}. Thus we may apply Lemma 3.8.2 inductively to establish the injectivity of the composite of the first two arrows of (3.15) and thereby complete the proof of the theorem.

Proof of Lemma 3.8.2. It will be convenient to bring the residue field T_{M}/𝔪 inside the coefficients of cohomology. To do this, we choose a finite field k containing T_{M}/𝔪 and tensor the source and target of (3.17) with k over F_{p} to obtain a map of k ⊗_{F_{p}} T_{M}/𝔪-modules. The chosen inclusion of T_{M}/𝔪 in k determines a projection

(3.18)  k ⊗_{F_{p}} T_{M}/𝔪 → k,

which realizes k as a local factor of k ⊗_{F_{p}} T_{M}/𝔪. Projecting onto this local factor, we recover our original map (3.17), but rewritten as

(3.19)  H^{1}(Mp; k)^{ord}[𝔪_{k}] → H^{1}(Np; k)^{ord}[𝔪′_{k}].

Here we regard the source as a W (k)⊗_{Z_{p}} T_{M}-module and the target as a W (k)⊗_{Z_{p}} T′_{N}-module. Also, we have written 𝔪_{k} to denote the the maximal ideal of W (k)⊗_{Z_{p}} T_{M} that is the kernel of the composite

W (k) ⊗_{Z_{p}} T_{M} → k ⊗_{F_{p}} T_{M}/𝔪 → k

(where the second arrow is given by the projection (3.18)) and 𝔪′_{k} to denote the maximal ideal of W (k) ⊗_{Z_{p}} T′_{N} that is the kernel of the composite

W (k) ⊗_{Z_{p}} T′_{N} → k ⊗_{F_{p}} T′_{N}/𝔪′ → k ⊗_{F_{p}} T_{M}/𝔪 → k

(where the second arrow is obtained by tensoring the injection T′_{N}/𝔪′ → T_{M}/𝔪 by k over F_{p} and the third arrow is given by the projection (3.18)). We must show that the map (3.19) is injective.

Our argument will rely on the results of [37, §2.2] which extend a well-known result of Ihara. Recall that the standard practice for analyzing the map (3.19) is to factor the map ϵ_{ℓ}^{∗} and so to write (3.19) as a composite

H^{1}(Mp; k)^{ord}[𝔪_{k}] ⟶^{α_{ℓ}} (H^{1}(Mp; k)^{ord}[𝔪_{k}])^{n_{ℓ}+1} ⟶^{β_{ℓ}} H^{1}(Np; k)^{ord}[𝔪′_{k}],

where

α_{ℓ} = \begin{cases}(1, −ℓ^{−1}T_{ℓ}) & if n_{ℓ} = 1 \\ (1, −ℓ^{−1}T_{ℓ}, ℓ^{−3}⟨ℓ⟩_{Mp}) & if n_{ℓ} = 2 \end{cases}

and

β_{ℓ} = \begin{cases}B_{ℓ,1}^{∗}π_{1} + B_{ℓ,ℓ}^{∗}π_{2} & if n_{ℓ} = 1 \\ B_{ℓ^{2},1}^{∗}π_{1} + B_{ℓ^{2},ℓ}^{∗}π_{2} + B_{ℓ^{2},ℓ^{2}}^{∗}π_{3} & if n_{ℓ} = 2 \end{cases}

with π_{i} denoting projection onto the ith factor of the product

(H^{1}(Np; k)^{ord}[𝔪_{k}])^{n_{ℓ}+1}.

The map α_{ℓ} is manifestly injective and, in the case n_{ℓ} = 2, Wiles has shown that the map β_{ℓ} is also injective. (See in particular the discussion at the top of [37, p. 497] or the discussion of Wiles’ results provided by [7, §§4.3] and [3, §§4.4, 4.5].) Thus when n_{ℓ} = 2 the lemma is proved.

However, if n_{ℓ} = 1, then the situation is more complicated. Lemma 2.5 of [37] provides an exact sequence

H^{1}((M/ℓ)p; k)_{𝔪′_{k}}^{ord} ⟶^{γ_{ℓ}} (H^{1}(Mp; k)_{𝔪′_{k}}^{ord})^{2} ⟶^{β_{ℓ}} H^{1}(Np; k)_{𝔪′_{k}}^{ord},

where γ_{ℓ} = (B_{ℓ,ℓ}^{∗}, −B_{ℓ,1}^{∗}). (Here the subscript 𝔪′_{k} denotes that we have localized at this maximal ideal.)

Let x ∈ H^{1}(Mp; k)[𝔪_{k}] be an element in the kernel of (3.19). Then (3.8) shows that we may find y ∈ H^{1}((M/ℓ)p; k) such that (x, −T_{ℓ}x) = (B_{ℓ,ℓ}^{∗}y, −B_{ℓ,1}^{∗}y). In particular, B_{ℓ,ℓ}^{∗}y = x is annihilated by 𝔪_{k} and so is an eigenvector for the full Hecke algebra T_{M}. The following result shows that B_{ℓ,ℓ}^{∗}y = 0 and hence that x = 0. This completes the proof of the lemma. □

Lemma 3.8.3. Let D be a natural number prime to p, let ℓ be a prime distinct from p and consider the map B_{ℓ,ℓ}^{∗} : H^{1}(Dp; F̄_{p})^{ord} → H^{1}(Dℓp; F̄_{p})^{ord}. If y is a class in the domain with the property that B_{ℓ,ℓ}^{∗}y is an eigenvector for the full Hecke ring T_{Dℓ}, corresponding to a maximal ideal for which the attached Galois representation into GL_{2}(F̄_{p}) is irreducible and p-distinguished, then B_{ℓ,ℓ}^{∗}y = 0.

Proof. We will prove the lemma by comparing the map B_{ℓ,ℓ}^{∗} on cohomology classes with the corresponding map on modular forms mod p. For such modular forms, the analogue of the lemma follows immediately from a consideration of q-expansions. For the comparison with modular forms mod p, we follow the discussion in the proof of [37, Thm. 2.1].

Let us make some initial reductions in the situation of the lemma. We let 𝔪 denote the maximal ideal in F̄_{p} ⊗_{F_{p}} T_{Dℓ} describing the action of the Hecke operators on the eigenvector B_{ℓ,ℓ}^{∗}y. Let 𝔪′ denote the intersection of 𝔪 with the subring F̄_{p} ⊗_{F_{p}} T′_{Dℓ} of F̄_{p} ⊗_{F_{p}} T_{Dℓ}. We may regard each of H^{1}(Dp; F̄_{p})^{ord} and H^{1}(Dℓp; F̄_{p})^{ord} as F̄_{p} ⊗_{F_{p}} T′_{Dℓ}-modules (the former via the natural map F̄_{p} ⊗_{F_{p}} T′_{Dℓ} → F̄_{p} ⊗_{F_{p}} T′_{D}) and the map B_{ℓ,ℓ}^{∗} is F̄_{p} ⊗_{F_{p}} T′_{Dℓ}-linear. If y_{𝔪′} denotes the projection of y onto the localization H^{1}(Dp; F̄_{p})_{𝔪′}^{ord} of H^{1}(Dp; F̄_{p})^{ord} at 𝔪′, then B_{ℓ,ℓ}^{∗}y_{𝔪′} = B_{ℓ,ℓ}^{∗}y (since B_{ℓ,ℓ}^{∗}y is assumed to be annihilated by 𝔪 and so in particular by 𝔪′). Also, there is a natural isomorphism

H^{1}(Dp; F̄_{p})_{𝔪′}^{ord} ≅ ∏_{𝔫⊃𝔪′(F̄_{p}⊗_{Fp} T_{D})} H^{1}(Dp; F̄_{p})_{𝔫}^{ord},

where 𝔫 ranges over all maximal ideals of F̄_{p} ⊗_{F_{p}} T_{D} containing 𝔪′(F̄_{p} ⊗_{F_{p}} T_{D}). Thus to show that B_{ℓ,ℓ}^{∗}y = 0, it suffices to show that B_{ℓ,ℓ}^{∗}y_{𝔫} = 0 for each such maximal ideal 𝔫, where y_{𝔫} denotes the projection of y onto H^{1}(Dp; F̄_{p})_{𝔫}^{ord}. Thus for the duration of the proof we assume that y ∈ H^{1}(Dp; F̄_{p})_{𝔫}^{ord}, for some maximal ideal 𝔫 of F̄_{p} ⊗_{F_{p}} T_{D} lying over 𝔪′.

As in the proof of [37, Thm. 2.1], we consider two cases: that in which the mod p diamond operators have non-trivial image in (F̄_{p} ⊗_{F_{p}} T_{Dℓ})/𝔪 and that in which they have trivial image.

In order to treat the first case, we begin by noting that the commutative diagram

(3.20)  \xymatrix{H^{1}(Dp; F̄_{p})_{𝔫}^{ord} \ar[r]^{B_{ℓ,ℓ}^{∗}} \ar[d]^{≅} & H^{1}(Dℓp; F̄_{p})_{𝔪}^{ord} \ar[d]^{≅} \\ (F̄_{p} ⊗_{F_{p}} J_{1}(Dp)[p]^{ord})_{𝔫} \ar[r]^{B_{ℓ,ℓ}^{∗}} & (F̄_{p} ⊗_{F_{p}} J_{1}(Dℓp)[p]^{ord})_{𝔪},}

in which the lower horizontal arrow is induced by the map of Jacobians arising by Picard functoriality applied to the degeneracy map B_{ℓ,ℓ} : X_{1}(Dℓp) → X_{1}(Dp), allows us to replace curves with coefficients in F̄_{p} with a consideration of p-torsion in the corresponding Jacobians (tensored by F̄_{p} over F_{p}). (The superscript ord on J_{1}(Dp)[p] and J_{1}(Dℓp)[pℓ] denotes the localization of these p-torsion modules at the p-ordinary part of the Hecke algebra, or equivalently, the image of the p-torsion modules under Hida’s idempotent e^{ord}.)

Let ρ̄ : G_{Q} → GL_{2}(F̄_{p}) denote the residual Galois representation attached to each of 𝔪 and 𝔫. From [37, Thm. 2.1] and its proof we conclude that there are isomorphisms of Galois modules

(F̄_{p} ⊗_{F_{p}} T_{D})_{𝔫}/p(F̄_{p} ⊗_{F_{p}} T_{D})_{𝔫} ⊗F̄_{p} ρ̄ ≅ (F̄_{p} ⊗_{F_{p}} J_{1}(Dp)[p]^{ord})_{𝔫}

and

(F̄_{p} ⊗_{F_{p}} T_{Dℓ})_{𝔪}/p(F̄_{p} ⊗_{F_{p}} T_{Dℓ})_{𝔪} ⊗F̄_{p} ρ̄ ≅ (F̄_{p} ⊗_{F_{p}} J_{1}(Dℓp)[p]^{ord})_{𝔪}.

Also, passing to p-torsion in the short exact sequence [37, (2.2)] for each of the maximal ideals 𝔪 and 𝔫 yields short exact sequences

0 → (F̄_{p} ⊗_{F_{p}} J_{1}(Dp)[p]^{ord})_{𝔫}^{0} → (F̄_{p} ⊗_{F_{p}} J_{1}(Dp)[p]^{ord})_{𝔫} →

(F̄_{p} ⊗_{F_{p}} J_{1}(Dp)[p]^{ord})_{𝔫}^{E} → 0

and

0 → (F̄_{p} ⊗_{F_{p}} J_{1}(Dℓp)[p]^{ord})_{𝔪}^{0} → (F̄_{p} ⊗_{F_{p}} J_{1}(Dℓp)[p]^{ord})_{𝔪} →

(F̄_{p} ⊗_{F_{p}} J_{1}(Dℓp)[p]^{ord})_{𝔪}^{E} → 0.

The lower horizontal arrow B_{ℓ,ℓ}^{∗} of (3.20) is G_{Q}-equivariant and also induces a morphism between these short-exact sequences.

If B_{ℓ,ℓ}^{∗}y is non-zero, then, since ρ̄ is an irreducible G_{Q}-module, we may find σ ∈ G_{Q} such that σ(B_{ℓ,ℓ}^{∗}y) has non-zero image in (F̄_{p} ⊗_{F_{p}} J_{1}(Dℓp)[p]^{ord})_{𝔪}^{E}. Replacing y by the image of σ(y) in (F̄_{p} ⊗_{F_{p}} J_{1}(Dp)[p]^{ord})_{𝔫}^{E}, we are reduced to proving the following claim.

Claim: If y ∈ (F̄_{p} ⊗_{F_{p}} J_{1}(Dp)[p]^{ord})_{𝔫}^{E} is such that B_{ℓ,ℓ}^{∗}y ∈ (F̄_{p} ⊗_{F_{p}} J_{1}(Dℓp)[p]^{ord})_{𝔫}^{E} is annihilated by 𝔪, then B_{ℓ,ℓ}^{∗}y vanishes.

Passing to the special fiber of the Néron model of each of J_{1}(Dp) and J_{1}(Dℓp) over Z_{p}[ζ_{p}] (where ζ_{p} denotes a primitive pth root of unity), the discussion in the proof of [37, Thm. 2.1] yields a commutative diagram in which the horizontal arrows are isomorphisms:

\xymatrix{(F̄_{p} ⊗_{F_{p}} J_{1}(Dp)[p]^{ord})_{𝔫}^{E} \ar[d]^{B_{ℓ,ℓ}^{∗}} \ar[r] & (H^{0}(Σ_{1}(D)^{μ}, Ω^{1}) ⊕ H^{0}(Σ_{1}(D)^{ét}, Ω^{1}))_{𝔫}^{ord} \ar[d]^{B_{ℓ,ℓ}^{∗}} \\ (F̄_{p} ⊗_{F_{p}} J_{1}(Dℓp)[p]^{ord})_{𝔪}^{E} \ar[r] & (H^{0}(Σ_{1}(Dℓ)^{μ}, Ω^{1}) ⊕ H^{0}(Σ_{1}(Dℓ)^{ét}, Ω^{1}))_{𝔪}^{ord} .}

(For any M prime to p, we denote by Σ_{1}(M)^{μ} and Σ_{1}(M)^{ét} the base-change from F_{p} to F̄_{p} of the two components of the special fiber of the canonical model of X_{1}(Mp) over Z_{p}[ζ_{p}].) Thus to prove the claim, it suffices to show that if z is a nonzero element of (H^{0}(Σ_{1}(D)^{μ}, Ω^{1}) ⊕ H^{0}(Σ_{1}(D)^{ét}, Ω^{1}))_{𝔫}^{ord}, then B_{ℓ,ℓ}^{∗} cannot lie in

the 𝔪-eigenspace of (H^{0}(Σ_{1}(Dℓ)^{μ}, Ω^{1}) ⊕ H^{0}(Σ_{1}(Dℓ)^{ét}, Ω^{1}))_{𝔪}^{ord} . This follows from the q-expansion principal; more precisely, the q-expansion of B_{ℓ,ℓ}^{∗}z at the cusp ∞ involves only powers of q^{ℓ} and so cannot be an eigenform for the full Hecke algebra T_{Dℓ}.

The second case (when the mod p diamond operators are trivial modulo 𝔪) is treated similarly, using the corresponding results from the proof of [37, Thm. 2.1].

□

4. Algebraic Iwasawa invariants

4.1. Selmer groups of modular forms. Let f = ∑ a_{n}q^{n} be a p-ordinary and p-stabilized newform of weight k ≥ 2, tame level N and character χ. Let K denote the finite extension of Q_{p} generated by the Fourier coefficients of f and let O denote the ring of integers of K; we write k for the residue field and fix also a uniformizer π of O. Let

ρ_{f} : G_{Q} → GL_{2}(K)

be the corresponding Galois representation, characterized by the fact that the characteristic polynomial under ρ_{f} of an arithmetic Frobenius at a prime ℓ ∤ Np is

X^{2} − a_{ℓ}X + χ(ℓ)ℓ^{k−1}.

By [14, Thm. 2.6] the restriction of ρ_{f} to G_{p} is of the form

(4.1)  ρ|_{G_{p}} ≅ ( \begin{matrix}ε^{k−1}χφ^{−1} & ∗ \\ 0 & φ \end{matrix} )

with φ : G_{p} → O^{×} the unramified character sending an arithmetic Frobenius to a_{p}. We assume that the semisimple residual representation

ρ̄_{f} : G_{Q} → GL_{2}(k)

is absolutely irreducible. It follows that, up to conjugation by GL_{2}(O), there is a unique integral model

ρ_{f} : G_{Q} → GL_{2}(O)

of ρ_{f}, which we now fix. For 0 ≤ i ≤ p − 2, let A_{f,i} denote a cofree O-module of corank 2 with G_{Q}-action via ρ_{f} ⊗ ω^{i}. We obtain from (4.1) and [16, Prop. 12.1] an O[G_{p}]-equivariant exact sequence

(4.2)  0 → (K/O)(ε^{k−1}χω^{i}φ^{−1}) → A_{f,i} → (K/O)(ω^{i}φ) → 0.

We write A′_{f,i} (resp. A′′_{f,i}) for the submodule (resp. quotient module) of A_{f,i} in the above sequence.

For a place v of Q_{∞} define

H_{s}^{1}(Q_{∞,v}, A_{f,i}) = \begin{cases}H^{1}(Q_{∞,v}, A_{f,i}) & v ≠ v_{p}; \\ im(H^{1}(Q_{∞,v_{p}}, A_{f,i}) → H^{1}(I_{v_{p}}, A′′_{f,i})) & v = v_{p}. \end{cases}

Following [10], the Selmer group of A_{f,i} is defined by

Sel(Q_{∞}, A_{f,i}) = ker(H^{1}(Q_{∞}, A_{f,i}) → ∏_{v} H_{s}^{1}(Q_{∞,v}, A_{f,i}))

= ker(H^{1}(Q_{Σ}/Q_{∞}, A_{f,i}) → ∏_{v∈Σ} H_{s}^{1}(Q_{∞,v}, A_{f,i}))

for any finite set of places Σ containing v_{p}, all archimedean places and all places dividing N. We regard Sel(Q_{∞}, A_{f,i}) as a Λ_{O}-module via the natural action of Γ.

Recall that μ^{alg}(f, ω^{i}) (resp. λ^{alg}(f, ω^{i})) is defined to be the largest power of π dividing (resp. the number of zeroes of) the characteristic power series of the Λ_{O}-dual of Sel(Q_{∞}, A_{f,i}) (assuming that this Selmer group is Λ_{O}-cotorsion).

Theorem 4.1.1. Let f be a p-ordinary and p-stabilized newform with ρ̄_{f} absolutely irreducible. Then Sel(Q_{∞}, A_{f,i}) is co-finitely generated, Λ_{O}-cotorsion and has no proper Λ_{O}-submodules of finite index. Furthermore, μ^{alg}(f, ω^{i}) vanishes if and only if Sel(Q_{∞}, A_{f,i})[π] is finite. If this is the case, then Sel(Q_{∞}, A_{f,i}) is O-divisible and

λ^{alg}(f, ω^{i}) = dim_{k} Sel(Q_{∞}, A_{f,i})[π].

Proof. It is shown in [10, Prop. 6] that Selmer groups are always co-finitely generated. The fact that they are also Λ_{O}-cotorsion for modular forms is proven in [20]. The equivalence of the vanishing of μ^{alg}(f, ω^{i}) and the finiteness of Sel(Q_{∞}, A_{f,i})[π] is now an immediate consequence of the structure theory of Λ_{O}-modules.

The proof of [13, Prop. 4.14] easily adapts to show that Sel(Q_{∞}, A_{f,i}) has no proper Λ_{O}-submodules of finite index. (As A_{f,i} need not be self-dual, this also requires the fact that Sel(Q_{∞}, A_{f,i}^{∗}) is Λ_{O}-cotorsion, which follows since A_{f,i}^{∗} is also modular.) When Sel(Q_{∞}, A_{f,i})[π] is finite, the maximal O-divisible Λ_{O}-submodule of Sel(Q_{∞}, A_{f,i}) has finite index, so that it must coincide with Sel(Q_{∞}, A_{f,i}); that is, Sel(Q_{∞}, A_{f,i}) is divisible. It now follows again from the structure theory of Λ_{O}-modules that as O-modules

Sel(Q_{∞}, A_{f,i}) ≅ (K/O)^{λ^{alg}(f,ω^{i})},

which proves the last statement. □

We close this section with a useful result on the local invariants H^{0}(G_{v}, A_{f,i}) for places v dividing primes ℓ ≠ p. Let cond_{ℓ}(ρ̄_{f}) denote the exponent of the highest power of ℓ that divides the conductor of ρ̄_{f}.

Lemma 4.1.2. Let v be a place of Q_{∞} dividing a prime ℓ ≠ p. If cond_{ℓ}(ρ̄_{f}) = ord_{ℓ}(N), then H^{0}(G_{v}, A_{f,i}) is O-divisible for all i.

Proof. By the invariance of the Swan conductor under reduction (see [23, §1]), we have

cond_{ℓ}(ρ_{f}) − cond_{ℓ}(ρ̄_{f}) = dim_{k} A_{f}[π]^{I_{ℓ}} − dim_{K} V_{f}^{I_{ℓ}}

where V_{f} is a two dimensional K-vector space with G_{Q}-action via ρ_{f}. Since cond_{ℓ}(ρ_{f}) equals ord_{ℓ}(N) by [1], we see that the hypothesis of the lemma is equivalent to the equality

dim_{k} A_{f}[π]^{I_{ℓ}} = dim_{K} V_{f}^{I_{ℓ}}.

It follows easily from this that A_{f}^{I_{ℓ}} = A_{f}^{I_{v}} is O-divisible. As the G_{v}/I_{v}-invariants of an O-divisible G_{v}/I_{v}-module are again O-divisible, we conclude that H^{0}(G_{v}, A_{f}) is divisible, as claimed. Since ω is unramified at ℓ, the above argument works for A_{f,i} as well. □

4.2. Residual Selmer groups. Let Ā denote a two-dimensional k-vector space equipped with a continuous k-linear action of G_{Q} and a k[G_{p}]-equivariant exact sequence

(4.3)  0 → Ā′ → Ā → Ā′′ → 0

with Ā′ and Ā′′ one-dimensional. We make the following assumptions on this data. (1) Ā is absolutely irreducible as a k[G_{Q}]-module;

(2) The k-vector space Ā′′ is unramified as a G_{p}-module;

(3) Ā is p-distinguished in the sense that the representation of G_{p} on Ā is non-scalar;

(4) Ā is modular: there exists a totally ramified extension K′ of the field of fractions of the Witt vectors of k, a p-stabilized newform f ∈ O′[[q]] of weight k ≥ 2 and a k[G_{Q}]-isomorphism Ā ≅ A_{f}[π′] identifying Ā′ with A′_{f}[π′] in the notation of the previous section; here π′ is a uniformizer of the ring of integers O′ of K′.

We will study Selmer groups of Ā and its cyclotomic twists Ā ⊗ ω^{i} with respect to various local conditions.

Definition 4.2.1. A finite/singular structure S on Ā⊗ω^{i} is a choice of k-subspaces

(4.4)  H_{f,S}^{1}(Q_{∞,v}, Ā ⊗ ω^{i}) ↪ H^{1}(Q_{∞,v}, Ā ⊗ ω^{i})

for each place v of Q_{∞}, subject to the restrictions:

(1) H_{f,S}^{1}(Q_{∞,v}, Ā ⊗ ω^{i}) = 0 for almost all v;

(2) H_{f,S}^{1}(Q_{∞,v_{p}}, Ā ⊗ ω^{i}) = ker(H^{1}(Q_{∞,v_{p}}, Ā ⊗ ω^{i}) → H^{1}(I_{v_{p}}, Ā′′ ⊗ ω^{i})).

(Note that we are not allowing any variation in the choice of condition at v_{p}). We define H_{s,S}^{1}(Q_{∞,v}, Ā ⊗ ω^{i}) as the cokernel of (4.4). The S-Selmer group of Ā ⊗ ω^{i} is

Sel_{S}(Q_{∞}, Ā ⊗ ω^{i}) = ker(H^{1}(Q_{∞}, Ā ⊗ ω^{i}) → ∏_{v} H_{s,S}^{1}(Q_{∞,v}, Ā ⊗ ω^{i}))

= ker(H^{1}(Q_{Σ}/Q_{∞}, Ā ⊗ ω^{i}) → ∏_{v∈Σ} H_{s,S}^{1}(Q_{∞,v}, Ā ⊗ ω^{i}))

for any finite set of places Σ containing p, all archimedean places, all places at which Ā is ramified and all places for which H_{f,S}^{1}(Q_{∞,v}, Ā ⊗ ω^{i}) does not vanish.

We will be especially interested in two kinds of finite/singular structures. First, the minimal structure S_{min} on Ā ⊗ ω^{i} is given by

H_{f,S_{min}}^{1}(Q_{∞,v}, Ā ⊗ ω^{i}) = 0

for v ≠ v_{p}.

Next let f be a newform as in (4) above. We define the induced structure S(f, i) on Ā ⊗ ω^{i} by setting

H_{f,S(f,i)}^{1}(Q_{∞,v}, Ā ⊗ ω^{i}) = ker(H^{1}(Q_{∞,v}, Ā ⊗ ω^{i}) → H_{s}^{1}(Q_{∞,v}, A_{f,i}))

for all places v of Q_{∞}. Note that by the definition of H_{s}^{1}(Q_{∞,v_{p}}, A_{f,i}) we have that H_{f,S(f,i)}^{1}(Q_{∞,v_{p}}, Ā ⊗ ω^{i}) is the kernel of the composite map

H^{1}(Q_{∞,v_{p}}, Ā ⊗ ω^{i}) → H^{1}(I_{v_{p}}, Ā′′ ⊗ ω^{i}) → H^{1}(I_{v_{p}}, A′′_{f,i});

in fact, since H^{0}(I_{v_{p}}, A′′_{f,i}) is either A′′_{f} (for i = 0) or else zero (for i ≠ 0), the latter map is injective, so that

(4.5) H_{f,S(f,i)}^{1}(Q_{∞,v_{p}}, Ā ⊗ ω^{i}) = ker(H^{1}(Q_{∞,v_{p}}, Ā ⊗ ω^{i}) → H^{1}(I_{v_{p}}, Ā′′ ⊗ ω^{i}))

as required. Lemma 4.2.2. For v ∤ p we have

H_{f,S(f,i)}^{1}(Q_{∞,v}, Ā ⊗ ω^{i}) = im(A_{f,i}^{G_{v}}/π ↪ H^{1}(Q_{∞,v}, Ā ⊗ ω^{i}))

In particular, H_{f,S(f,i)}^{1}(Q_{∞,v}, Ā ⊗ ω^{i}) = 0 if Ā ⊗ ω^{i} is unramified at v ≠ v_{p} so that S(f, i) is a finite/singular structure. The natural map

Sel_{S(f,i)}(Q_{∞}, Ā ⊗ ω^{i}) → Sel(Q_{∞}, A_{f,i})[π]

is an isomorphism.

Proof. The lemma follows from various exact sequences in cohomology coming from the exact sequence

0 → Ā ⊗ ω^{i} → A_{f,i} ⟶^{π} A_{f,i} → 0

together with Lemma 4.1.2. We leave the details to the reader. □

Proposition 4.2.3. Let Ā be as above. Then there exists a newform f as in (4) above such that the finite/singular structures S_{min} and S(A_{f,i}) coincide under the isomorphism Ā ⊗ ω^{i} ≅ A_{f,i}[π′] for any i.

Proof. Let N denote the tame conductor of the Galois representation Ā. As f is p-distinguished, by [5, Thm. 6.4] there exists a finite totally ramified extension K′ of K and a (not necessarily unique) p-stabilized newform f ∈ K′[[q]] of tame level N and weight 2 satisfying the condition of (4). (The final condition in (4) is in fact already automatic from the more standard level lowering result [5, Thm. 1.1] unless Ā is an unramified G_{p}-module, in which case the Selmer case of [5, Thm. 6.4] ensures the existence of such an f.)

Since the tame conductor of f equals the conductor of Ā, by Lemma 4.1.2, A_{f,i}^{G_{v}} is divisible for any place v ≠ v_{p}. It then follows from Lemma 4.2.2 that the structure S(A_{f,i}) on A_{f,i}[π′] ≅ Ā agrees with S_{min}, as desired. □

The next proposition, which follows from a result of Greenberg, is crucial to our method.

Proposition 4.2.4. For f be as above there is an exact sequence

0 → Sel_{S(f,i)}(Q_{∞}, Ā ⊗ ω^{i}) → H^{1}(Q_{Σ}/Q_{∞}, Ā ⊗ ω^{i}) →

∏_{v∈Σ} H_{s,S(f,i)}^{1}(Q_{∞,v}, Ā ⊗ ω^{i}) → 0

for any finite set of places Σ containing v_{p}, all archimedean places and all places dividing the tame level of f.

Proof. Since A_{f,i} is odd and Sel(Q_{∞}, A_{f,i}) is Λ_{O}-cotorsion, by [15, Prop. 2.1] there is an exact sequence

0 → Sel(Q_{∞}, A_{f,i}) → H^{1}(Q_{Σ}/Q_{∞}, A_{f,i}) → ∏_{v∈Σ} H_{s}^{1}(Q_{∞,v}, A_{f,i}) → 0

with Σ as above. As Sel(Q_{∞}, A_{f,i}) is O-divisible, the π-torsion of this sequence is an exact sequence

0 → Sel(Q_{∞}, A_{f,i})[π] → H^{1}(Q_{Σ}/Q_{∞}, A_{f,i})[π] → ∏_{v∈Σ} H_{s}^{1}(Q_{∞,v}, A_{f,i})[π] → 0.

By Lemma 4.2.2 and the definition of the induced structure, we see that this sequence identifies with that of the proposition. (Note that (4.5) shows that the local conditions agree at the place v_{p}.) □

Our main algebraic theorems are consequences of the next result.

Corollary 4.2.5. Let Ā be as above and let f be a p-stabilized newform of tame level N_{f} such that A_{f}[π] ≅ Ā as in (4). Then the sequence

(4.6)  0 → Sel_{S_{min}}(Q_{∞}, Ā ⊗ ω^{i}) → Sel(Q_{∞}, A_{f,i})[π] → ∏_{v|N_{f}} A_{f,i}^{G_{v}}/π → 0

is exact for any i.

Proof. The exactness of (4.6) follows from the definitions and Lemma 4.2.2 except for the surjectivity of the last map. For this, by Proposition 4.2.3 there exists a totally ramified extension K′/K and a p-stabilized newform f_{0} over K′ such that (Ā ⊗ ω^{i}, S_{min}) ≅ (A_{f_{0},i}[π′], S(f_{0}, i)). The surjectivity is then a formal consequence of the exact sequence of Proposition 4.2.4 applied to both A_{f_{0},i} and A_{f,i} with Σ = {v | N_{f}p}. □

4.3. Algebraic Iwasawa invariants. Let Ā be as in the previous section. We write ρ̄ : G_{Q} → GL_{2}(k) for the corresponding Galois representation. We now use Corollary 4.2.5 to study the relations between Selmer groups of newforms in the Hida family of ρ̄. We first consider the behavior of the corresponding finite/singular structures on Ā.

Lemma 4.3.1. Let f be a newform on the branch T(𝔞) of the Hida family of ρ̄. Then for any place v, the dimension of H_{f,S(f,i)}^{1}(Q_{∞,v}, Ā⊗ ω^{i}) depends only on the branch T(𝔞) and i.

Remark 4.3.2. It was pointed out to the authors by Ralph Greenberg that although the dimension of the local condition is constant along branches of Hida families, it is possible (when the branch is special at v) that the actual subspace H_{f,S(f,i)}^{1}(Q_{∞,v}, Ā ⊗ ω^{i}) varies as f varies along the branch T(𝔞). It would be interesting to determine if such an example actually exists.

Proof. Fix a place v ≠ v_{p} dividing the rational prime ℓ. Consider first the unramified G_{ℓ}-representation (V_{f})_{I_{ℓ}}. By the proof of Proposition 2.2.3 the dimension of this space is constant on the branch T(𝔞). It follows that, as a G_{ℓ}-representation, the residual representation of (V_{f})_{I_{ℓ}} is independent of f on T(𝔞).

Consider now the exact sequence

(4.7)  0 → A_{f,i}^{G_{v}}/π → H^{1}(Q_{∞,v}, Ā ⊗ ω^{i}) → H^{1}(Q_{∞,v}, A_{f,i})[π] → 0.

By [15, Prop. 2.4], H^{1}(Q_{∞,v}, A_{f,i})[π] has k-dimension equal to the multiplicity of ω^{1−i} in the residual representation of the unramified G_{ℓ}-representation (V_{f})_{I_{ℓ}}. By the above discussion this is independent of f on the branch T(𝔞), as claimed. □

For a branch T(𝔞) of the Hida family of ρ̄ and a place v ≠ v_{p}, we may now define

δ_{v}(𝔞, ω^{i}) := dim_{k} H_{f,S(f,i)}^{1}(Q_{∞,v}, Ā ⊗ ω^{i}) = dim_{k} A_{f,i}^{G_{v}}/π

for any form f lying on the branch T(𝔞).

We say that

μ^{alg}(ρ̄, ω^{i}) = 0

if Sel_{S_{min}}(Q_{∞}, Ā ⊗ ω^{i}) is finite dimensional over k; if this is the case we define

λ^{alg}(ρ̄, ω^{i}) = dim_{k} Sel_{S_{min}}(Q_{∞}, Ā ⊗ ω^{i}).

Theorem 4.3.3. Let ρ̄ be as above. Let f be any newform in the Hida family of ρ̄. Then μ^{alg}(ρ̄, ω^{i}) = 0 if and only if μ^{alg}(f, ω^{i}) = 0.

Proof. By Theorem 4.1.1, μ^{alg}(f, ω^{i}) vanishes if and only if Sel(Q_{∞}, A_{f,i})[π] is finite dimensional. By Corollary 4.2.5 this is equivalent to the finite dimensionality of Sel_{S_{min}}(Q_{∞}, Ā ⊗ ω^{i}), as claimed. □

Theorem 4.3.4. Let ρ̄ be as above. Assume that μ^{alg}(ρ̄, ω^{i}) = 0.

(1) Let f be a newform of level N_{f} lying on the branch T(𝔞) of the Hida family of ρ̄. Then

λ^{alg}(f, ω^{i}) = λ^{alg}(ρ̄, ω^{i}) + ∑_{v|N_{f}} δ_{v}(𝔞, ω^{i}).

In particular, λ^{alg}(f, ω^{i}) depends only on the branch T(𝔞) of f; we write λ^{alg}(𝔞, ω^{i}) for this value.

(2) Let T(𝔞_{1}) and T(𝔞_{2}) be two branches of the Hida family of ρ̄. Then

λ^{alg}(𝔞_{1}, ω^{i}) − λ^{alg}(𝔞_{2}, ω^{i}) = ∑_{v≠v_{p}} δ_{v}(𝔞_{1}, ω^{i}) − δ_{v}(𝔞_{2}, ω^{i}).

Proof. The first statement is immediate from Corollary 4.2.5 and the definition of δ_{v}(𝔞, ω^{i}). The second statement follows from the first. □

We remark that in Section 5 we will see that the formulas of the preceding theorem compare well with the formulas of Theorem 3.7.7.

5. Applications to the main conjecture and examples

5.1. The main conjecture. Let O be the ring of integers of a finite extension K of Q_{p} and let f ∈ O[[q]] be a p-ordinary eigenform such that the residual representation ρ̄_{f} is irreducible. Let L_{p}^{alg}(f, ω^{i}) ∈ Λ_{O} denote a generator of the characteristic power series of the Λ_{O}-dual of the Selmer group Sel(Q_{∞}, A_{f,i}) of Section 4.1. Let L_{p}^{an}(f, ω^{i}) ∈ Λ_{O} denote the usual p-adic L-function of f ⊗ ω^{i} (computed with respect to some canonical period). The main conjecture of Iwasawa theory in this context is the following; it is independent of the particular choice of coefficient field K over which f is defined.

Conjecture 5.1.1. There is a unit u ∈ Λ_{O}^{×} such that

L_{p}^{alg}(f, ω^{i}) · u = L_{p}^{an}(f, ω^{i}).

We remark that the Selmer group in the above conjecture is not the Bloch–Kato Selmer group in which the local condition at p is defined via crystalline periods, but instead is the Greenberg Selmer group in which the local condition at p is defined via the ordinary filtration as in Section 4.1. The former group is always contained in the latter group and the quotient is trivial unless the analytic p-adic L-function has a trivial zero, in which case it has corank one (see [10, pp. 108-109]). It is for this reason that in our statement of the main conjecture we do not need to consider separately the case of trivial zeroes.

The following deep theorem of Kato [20] establishes one divisibility of the main conjecture.

Theorem 5.1.2 (Kato). There is a u ∈ Λ_{O} ⊗ Q_{p} such that

L_{p}^{alg}(f, ω^{i}) · u = L_{p}^{an}(f, ω^{i}).

In particular, to verify the main conjecture for f ⊗ ω^{i} it suffices to check that

μ^{alg}(f, ω^{i}) = μ^{an}(f, ω^{i}) and λ^{alg}(f, ω^{i}) = λ^{an}(f, ω^{i}).

Our results in Sections 3 and 4 yield the following result.

Theorem 5.1.3. Let k be a finite field of characteristic p and let ρ̄: G_{Q} → GL_{2}(k) be an irreducible, modular, p-ordinary and p-distinguished representation; as always we fix a choice of p-stabilization. Suppose that

μ^{alg}(f_{0}, ω^{i}) = μ^{an}(f_{0}, ω^{i}) = 0 and λ^{alg}(f_{0}, ω^{i}) = λ^{an}(f_{0}, ω^{i})

for some f_{0} in the Hida family attached to ρ̄ and some i. Then

μ^{alg}(f, ω^{i}) = μ^{an}(f, ω^{i}) = 0 and λ^{alg}(f, ω^{i}) = λ^{an}(f, ω^{i})

for every f in the Hida family attached to ρ̄.

Before giving a proof, we first state an immediate corollary of Theorem 5.1.3 and Kato’s result.

Corollary 5.1.4. Let ρ̄ be as above and suppose that μ^{alg}(ρ̄, ω^{i}) = μ^{an}(ρ̄, ω^{i}) = 0 for some i. If the main conjecture holds for f_{0} ⊗ ω^{i} for one form f_{0} in the Hida family of ρ̄, then the main conjecture holds for f ⊗ ω^{i} for every form f in the Hida family of ρ̄.

Since every Hida family contains a form of weight two, this corollary in particular reduces the main conjecture to the case of weight two and to the conjecture on the vanishing of the μ-invariants.

The proof of Theorem 5.1.3 is based on the following lemma which relates the invariants e_{ℓ}(𝔞, ω^{i}) and δ_{v}(𝔞, ω^{i}) of Theorem 3.7.7 and Section 4.3.

Lemma 5.1.5. Let 𝔞_{1} and 𝔞_{2} be minimal primes of T_{Σ}. For any prime ℓ ≠ p

∑_{v|ℓ} δ_{v}(𝔞_{1}, ω^{i}) − δ_{v}(𝔞_{2}, ω^{i}) = e_{ℓ}(𝔞_{2}, ω^{i}) − e_{ℓ}(𝔞_{1}, ω^{i})

where the sum is taken over all primes v of Q_{∞} over ℓ.

Proof. To prove the lemma it suffices to see that for any minimal prime 𝔞 of T_{Σ}, the sum

e_{ℓ}(𝔞, ω^{i}) + ∑_{v|ℓ} δ_{v}(𝔞, ω^{i})

is independent of 𝔞. For this, fix a classical newform f on the branch T(𝔞). Consider first the group H^{1}(Q_{∞,v}, A_{f,i}). Since H^{1}(Q_{∞,v}, A_{f,i}) is divisible (as G_{v} has pcohomological dimension one) we have

dim_{k} H^{1}(Q_{∞,v}, A_{f,i})[π] = λ (char_{Λ_{O}}(H^{1}(Q_{∞,v}, A_{f,i})^{∨}))

where M^{∨} = Hom(M, Q_{p}/Z_{p}).

By [15, Proposition 2.4] we have

char_{Λ_{O}} (⊕_{v|ℓ}H^{1}(Q_{∞,v}, A_{f,i})^{∨}) = E_{ℓ}(f, ⟨ℓ⟩_{p,i}^{−1}) · Λ_{O}.

Since E_{ℓ}(f, ⟨ℓ⟩_{p,i}^{−1}) is simply E_{ℓ}(𝔞, ⟨ℓ⟩_{p,i}^{−1}) mod ℘_{f}, we conclude that

(5.1)  ∑_{v|ℓ} dim_{k} H^{1}(Q_{∞,v}, A_{f,i})[π] = e_{ℓ}(𝔞, ω^{i}).

Consider now the exact sequence

0 → A_{f,i}^{G_{v}}/π → H^{1}(Q_{∞,v}, Ā ⊗ ω^{i}) → H^{1}(Q_{∞,v}, A_{f,i})[π] → 0.

Since the first term has k-dimension δ_{v}(𝔞, ω^{i}) and the second term is certainly independent of 𝔞, the lemma now follows from (5.1). □

Proof of Theorem 5.1.3. Let f be a form in the Hida family of ρ̄. The vanishing of the μ-invariants of f is immediate from that for f_{0} and Theorems 4.3.3 and 3.7.5.

By Lemma 5.1.5 and Theorems 4.3.4 and 3.7.7 we see also that

λ^{alg}(f, ω^{i}) − λ^{alg}(f_{0}, ω^{i}) = λ^{an}(f, ω^{i}) − λ^{an}(f_{0}, ω^{i}).

Since we are assuming the main conjecture for f_{0} ⊗ ω^{i}, we have that

λ^{alg}(f_{0}, ω^{i}) = λ^{an}(f_{0}, ω^{i}),

so that it follows that

λ^{alg}(f, ω^{i}) = λ^{an}(f, ω^{i})

as desired. □

5.2. Raising the level. We conclude our general discussion with some results on branches of Hida families. It is well-known (see [8, 9]) which levels can occur among forms in the Hida family of ρ̄. In the next proposition we list the cases in which the invariant δ_{v}(·, ω^{i}) increases.

Proposition 5.2.1. Let ρ̄ : G_{Q} → GL_{2}(k) be an irreducible, p-ordinary and pdistinguished modular Galois representation; assume further that ρ̄ is ramified at p. Let Σ be some finite set of primes not containing p and let ℓ ≠ p be a prime at which ρ̄ is unramified. Set Σ′ = Σ ∪ {ℓ} and let T_{Σ}, T_{Σ′} be the Hida algebras associated to ρ̄. Let a_{ℓ} = Trace ρ̄(Frob_{ℓ}) and c_{ℓ} = det ρ̄(Frob_{ℓ}), both viewed as elements of k. Then for every branch T(𝔞) of T_{Σ}:

(1) If there is an i such that a_{ℓ} = ℓ^{−i} + ℓ^{1−i} and c_{ℓ} = ℓ^{1−2i}, then there is a branch T(𝔟) of T_{Σ′} such that

N(𝔟) = N(𝔞)ℓ and δ_{v}(𝔟, ω^{i}) = 1

for all places v dividing ℓ.

(2) If ℓ ≡ 1 (mod p) and a_{ℓ} = c_{ℓ} + 1, then there is a branch T(𝔟) of T_{Σ′} (distinct from the branch provided by (1)) such that

N(𝔟) = N(𝔞)ℓ and δ_{v}(𝔟, ω^{i}) = 1

for all places v dividing ℓ and all i.

(3) If ℓ ≡ 1 (mod p), a_{ℓ} = 2 and c_{ℓ} = 1, then there are distinct branches T(𝔟_{1}) and T(𝔟_{2}) of T_{Σ′} such that

N(𝔟_{1}) = N(𝔟_{2}) = N(𝔞)ℓ^{2} and δ_{v}(𝔟_{1}, ω^{i}) = δ_{v}(𝔟_{2}, ω^{i}) = 2

for all places v dividing ℓ and all i.

(4) If ℓ ≡ −1 (mod p), a_{ℓ} = 0 and c_{ℓ} = −1, then there is a branch T(𝔟) of T_{Σ′} such that

N(𝔟) = N(𝔞)ℓ^{2} and δ_{v}(𝔟, ω^{i}) = 1

for all places v dividing ℓ and all i.

Note in particular that if (3) holds, then (1) and (2) also hold, so that the proposition exhibits four distinct non-minimal branches of T′_{Σ} for each branch of T_{Σ}.

Proof. If (1) holds, then one sees easily that the eigenvalues of Frob_{ℓ} on ρ̄ equal ℓ^{1−i} and ℓ^{−i}. It follows that there exists a ramified representation

τ_{ℓ} : G_{ℓ} → GL_{2}(Z_{p})

such that τ̄_{ℓ} ⊗ k ≅ ρ̄|_{G_{ℓ}} and

τ_{ℓ} = ( \begin{matrix}ω^{1−i} & ∗ \\ 0 & ω^{−i} \end{matrix} ) .

By [9, Theorem 1] (which applies since ρ̄ is modular of weight 2 by [5, Theorem 1.1]) there exists a newform g of tame level Nℓ in the Hida family of ρ̄ such that ρ_{g}|_{I_{ℓ}} ≅ τ_{ℓ}|_{I_{ℓ}}. Since ρ̄ ⊗ ω^{i} is unramified at ℓ with Frobenius eigenvalues ℓ and 1, one computes easily that

δ_{v}(A_{g}, ω^{i}) = 1

for any place v dividing ℓ. In particular, the branch 𝔞_{1} on which g lies satisfies the requirements of the proposition.

The proofs for (2) (principal series), (3) (either special or principal series) and (4) (supercuspidal) are entirely similar; the conditions on ℓ, a_{ℓ} and c_{ℓ} are simply those obtained by combining the conditions of [8, p. 435] for ramified lifts of the appropriate type to exist with the requirement that ρ̄(Frob_{ℓ}) has a trivial eigenvalue. Note also that ω|_{G_{ℓ}} is trivial (resp. quadratic) in (2) and (3) (resp. (4)), so that the choice of i is irrelevant. Finally, the branches in (1) and (2) (resp. in (3)) are distinct since one is special at ℓ and the other is principal series at ℓ. □

It would be interesting to determine when the branches of the preceding proposition intersect. The following proposition uses Λ-adic level raising to give some insight into this question.

Proposition 5.2.2. We maintain the hypotheses of the previous proposition. Let 𝔞 be a minimal prime of T_{Σ} such that

(T_{ℓ} mod 𝔞′)^{2} − ℓ^{−2}⟨ℓ⟩_{N(𝔞)p}(ℓ + 1)^{2}

is not a unit in T(𝔞)^{∘}. (Recall that 𝔞′ is a minimal prime of T_{N(𝔞)}^{new} sitting over 𝔞 and that T(𝔞)^{∘} ≅ T_{N(𝔞)}^{new}/𝔞′.) Then there exists some minimal prime 𝔟 of T_{Σ′} with N(𝔟) = N(𝔞)ℓ and a height one prime ℘ of T_{Σ′} such that T(𝔞) and T(𝔟) cross at ℘. (That is, both 𝔟 and the pre-image of 𝔞 under the natural map T_{Σ′} → T_{Σ} are contained in ℘.)

Proof. Attached to the minimal prime 𝔞 we have the Λ-adic modular form

f(𝔞, q) = ∑_{n≥1}(T_{n} mod 𝔞′)q^{n}.

By [4, Theorem 6C], there exists a finite extension R of T(𝔞)^{∘}, a Λ-adic modular form g(q) = ∑_{n≥1} b_{n}q^{n} ∈ R[[q]] new of level N(𝔞)ℓ, and a height one prime ℘′ of R such that

(5.2)  (T_{n} mod 𝔞′) ≡ b_{n} (mod ℘′)

for each n relatively prime to N(𝔞)ℓ.

The Λ-adic form g(q) corresponds to some minimal prime ideal of T_{Σ}. Indeed, there is a map T_{N(𝔞)ℓ}^{new} → R that sends T_{n} to b_{n}. Let 𝔟′ ⊆ T_{N(𝔞)ℓ}^{new} denote the kernel of this map; it is necessarily a minimal prime since both the source and the target are finite extensions of Λ. The preimage of 𝔟′ in ∏_{M|N(Σ′)} T_{M}^{new} is a minimal prime and thus, by Proposition 2.3.2, corresponds to some minimal prime of T′_{N(Σ′)}. Since the residual representation attached to g(q) equals ρ̄, this minimal prime yields a minimal prime of T_{Σ′} which we denote by 𝔟. Note that by construction N(𝔟) = N(𝔞)ℓ.

We thus have a map T_{Σ′} → R (sending T_{n} to b_{n}) with kernel 𝔟. If let 𝔞̃ denotes the pre-image of 𝔞 in T_{Σ′}, then we also have a map T_{Σ′} → R with kernel 𝔞̃ (given by reducing mod 𝔞̃ and then embedding the image T(𝔞) into R). By (5.2), the reduction of these two maps modulo ℘′ are the same. If we let ℘ denote the kernel of either of these maps modulo ℘′, then ℘ is a height one prime containing both of 𝔟 and 𝔞̃, as desired. □

The ideal ℘ of the previous proposition could potentially be a prime ideal lying over the principal ideal (p). It would be interesting to determine whether or not these branches actually meet at a prime ideal of residue characteristic zero.

5.3. Examples.

Example 5.3.1. Set p = 11 and let f denote the weight 2 newform associated to the elliptic curve X_{0}(11). It follows from the fact that f is the only newform of weight 2 and level dividing 11 that T := T_{Σ}(ρ̄_{f}) ≅ Λ for Σ = ∅. Thus for k ≥ 2, there is a unique newform f_{k} of weight k and level dividing 11 that is congruent to f modulo 11. For example, f_{12} is the 11-ordinary, 11-stabilized oldform of level 11 attached to the Ramanujan Δ-function.

The p-adic L-function of this family was studied in detail in [14]. Also, in [11], the congruence between X_{0}(11) and Δ was exploited to gain information about the Selmer groups of both forms. We review below what is known in these examples and then go on to study the other branches of this Hida family with tame conductor greater than one.

We first verify the main conjecture for f. Since X_{0}(11) has split multiplicative reduction at p = 11, the p-adic L-function L_{p}^{an}(f, T) has a trivial zero at T = 0.

By the Mazur, Tate and Teitelbaum conjecture (proved by Greenberg and Stevens [14]) we have an explicit formula for the derivative of L_{p}^{an}(f, T) at 0; that is,

\frac{d}{dT} L_{p}^{an}(f, T)|_{T =0} = \frac{L_{p}(f)}{log_{p}(γ)} · \frac{L(f, 1)}{Ω_{f}}.

(Here, γ is a topological generator of 1+pZ_{p} that is implicitly chosen by writing the p-adic L-function in the T -variable; the appearance of the extra factor of log_{p}(γ) above is accounted for by the fact that we have written this formula in the T -variable rather than the more standard s-variable.) One checks that L_{p}(f) and log_{p}(γ) are exactly divisible by 11 while L(f, 1)/Ω_{f} is an 11-adic unit. Thus L_{p}^{an}(f, T) is a unit multiple of T so that λ^{an}(f) = 1 and μ^{an}(f) = 0.

Since L(f, 1)/Ω_{f} is an 11-adic unit, by an Euler system argument of Kolyvagin, the classical 11-adic Selmer group Sel_{p}(X_{0}(11)/Q) of the elliptic curve X_{0}(11) vanishes. It thus follows from a refined control theorem of Greenberg ([13, Proposition 3.7]) that Sel_{p}(X_{0}(11)/Q_{∞}) vanishes as well. The latter group agrees with the Bloch–Kato Selmer group of f; as f has a trivial zero, it follows that the Greenberg Selmer group Sel_{p}(Q_{∞}, A_{f}) is simply Q_{p}/Z_{p} with trivial Galois action. (This example is worked out in detail in [11, Example 3].) Thus L_{p}^{alg}(f, T) is a unit multiple of T, so that λ^{alg}(f) = 1 and μ^{alg}(f) = 0. In particular, this verifies the main conjecture for X_{0}(11) at p = 11.

Corollary 5.1.4 now shows that the main conjecture holds at p = 11 for each form f_{k} with k ≥ 2, and Theorem 5.1.3 shows that

λ^{an}(f_{k}) = λ^{alg}(f_{k}) = 1 for k ≥ 2.

Note that the p-adic L-function corresponding to X_{0}(11) is the only p-adic Lfunction in the Hida family with a trivial zero. Nonetheless, the existence of this trivial zero forces every p-adic L-function in the family to have at least one zero.

For k ≡ 2 (mod 10), the unique zero of L_{p}^{an}(f_{k}) can be explained by the functional equation of the p-adic L-function. Namely, the sign of this functional equation is −1 for such k and thus L_{p}^{an}(f_{k}) vanishes at the character that sends x to x^{\frac{k−2}{2}} . Using this observation, we can determine the two-variable p-adic L-function attached to ρ̄_{f} (which we denote by L(ρ̄_{f})). By Theorem 3.7.7, we have that λ(L(ρ̄_{f})) = 1 and hence, under the identification of (3.9),

L(ρ̄_{f}) = (T − a_{0}) · u(T)

with a_{0} ∈ T and u(T) ∈ T[[T]] a unit. Since L(ρ̄_{f}) specializes at weight k to L_{p}(f_{k}) (Proposition 3.4.3), we know that a_{0} specializes at weight k to the unique zero of L_{p}(f_{k}). In particular, for k ≡ 2 (mod 10), we have that a_{0} specializes to γ^{\frac{k−2}{2}} .

Thus, a_{0} = γ^{−1}⟨γ⟩p^{\frac{1}{2}} since this element has the correct specialization for infinitely many k.

So far we have only applied our results to minimal lifts of ρ̄_{f}; however, Corollary 5.1.4 applies to every modular form lifting this representation. For instance, we can use Proposition 5.2.1 to explicitly find primes ℓ to add to the level that will produce modular forms that have higher λ-invariants and for which we still know the main conjecture.

The first prime that satisfies condition (3) (and thus (1) and (2)) of Proposition 5.2.1 is ℓ = 1321. If we set Σ = {1321}, then T_{Σ} will have four distinct branches T(𝔞_{1}), …, T(𝔞_{4}) with λ(𝔞_{1}) = λ(𝔞_{2}) = 2 and λ(𝔞_{3}) = λ(𝔞_{4}) = 3. (Here and in what follows, we do not distinguish between the analytic and algebraic λ-invariants in cases where the main conjecture is known to be true.) Note that 1321 is inert in the cyclotomic Z_{11}-extension of Q and so there is a unique prime v of Q_{∞} sitting over 1321.

The prime ℓ′ = 2113 is also inert in Q_{∞}/Q and satisfies condition (3) of Proposition 5.2.1. If Σ′ = {1321, 2113}, we may apply Proposition 5.2.1 to each of the five branches we have exhibited in T_{Σ} to see that T_{Σ′} possesses at least 20 additional branches with λ-invariants ranging between 3 and 5.

Example 5.3.2. We conclude by considering an example of Greenberg and Vatsal in [15]. Consider the elliptic curves

E_{1} : y^{2} = x^{3} + x − 10 E_{2} : y^{2} = x^{3} − 584x + 5444,

of conductors 52 and 364 respectively, which are ordinary and congruent mod 5. One computes that λ^{an}(E_{1}) = μ^{an}(E_{1}) = 0 so that Kato’s divisibility (Theorem 5.1.2) yields the main conjecture for E_{1} at p = 5. The main conjecture for E_{2} at p = 5 follows and in this case we have λ(E_{2}) = 5. (This example is discussed in detail in [15, pp. 22, 44].)

We now examine this congruence from the point of view of Hida theory. Let f_{i} denote the newform of weight 2 associated to E_{i}. One checks that f_{1} is not congruent modulo 5 to any other modular forms of weight two and level dividing 2^{2} ·5·13^{2} (using [32], for example). It follows that we have T_{Σ} ≅ Λ for Σ = {2, 13}. For consistency of notation, let us denote the unique irreducible component of this space as T(𝔞_{1}). Moreover, since μ(f_{1}) = λ(f_{1}) = 0, the two-variable p-adic L-function attached to T_{Σ} is a unit and λ(𝔞_{1}) = 0.

The prime ℓ = 7 satisfies condition (1) of Proposition 5.2.1 (with i = 0). Thus, if Σ′ = {2, 7, 13}, then T_{Σ′} contains an irreducible component T(𝔞_{2}) not contained in T_{Σ}. Moreover, since f_{2} is the unique normalized newform of weight 2 congruent modulo 5 to f_{1} with level dividing 2^{2} · 5 · 7^{2} · 13^{2}, it follows that T_{Σ′} has rank 2 over Λ. Hence, T(𝔞_{2}) is the only branch of T_{Σ′} not coming from T_{Σ}. By Theorems 3.7.7 and 4.3.4, we have that λ(𝔞_{2}) = 5. Since f_{2} sits on the branch T(𝔞_{2}), it follows that λ(f_{2}) = 5.

We close with some questions. Proposition 5.2.2 establishes that the branches T(𝔞_{1}) and T(𝔞_{2}) must cross at some (non-classical) height one prime, but does not exclude the possibility that they cross at the prime (p). Do they in fact cross at a prime of residue characteristic zero and if so could one compute p-adic approximations of this prime? How many such crossing points do these two branches share? It appears at present that little is known about the shape of these Hida families when multiple branches appear (even in any particular case).

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(Matthew Emerton) Mathematics Department, Northwestern University, Evanston, IL

(Robert Pollack) Department of Mathematics, Boston University, Boston, MA

(Tom Weston) Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA

E-mail address, Matthew Emerton: emerton@math.northwestern.edu

E-mail address, Robert Pollack: rpollack@math.bu.edu E-mail address, Tom Weston: weston@math.umass.edu