APPENDIX: AN INTRODUCTION TO THE

p-ADIC GEOMETRY OF MODULAR CURVES

Matthew Emerton

University of Michigan

The aim of this appendix is to give some feeling for the ideas behind the p-adic theory of modular curves and modular forms (at a more down to earth level than that of [18] or [25], which are the standard references on this topic). It provides a written approximation to an informal lecture given on this topic by the author at the 1999 PCMI meeting. It was in the informal spirit of the lecture to make many assertions and allusions without providing details. That spirit may also permeate this written version, but some attempt has been made to give relevant references to the literature. In fact, although the bibliography is (of course and inevitably) incomplete, it does include many of the major papers in the field, and the final section is devoted to annotating briefly some of those references.

As one might guess, the background and sophistication that is assumed of the reader varies widely from section to section. A large part of the presentation is devoted to treating the case p = 2, where many phenomena can be discovered and investigated via explicit calculation; in these parts of the presentation we assume comparatively little background. On the other hand, in order to understand the theory behind the calculations, and the generalization to arbitrary primes p, more sophisticated ideas (such as the consideration of elliptic curves and modular forms defined over rings other than fields) cannot be avoided. Nevertheless, we have attempted to make the discussion as clear and intuitive as possible; we leave it to the reader and the references to fortify the intuition with correct mathematics, while the author will accept responsibility for any lack of clarity.

Acknowledgments: I would like to thank David Ben-Zvi, Mike Roth and Robert Coleman for some helpful comments on earlier drafts of the present work.

1. The curve X_{0}(2). All the essential ideas in the p-adic theory can already be seen in the case p = 2, and this case has the merit that one can easily perform explicit computations. We begin by describing some of these, working first over ℂ.

We let H^{∗} denote the extended upper half-plane, that is, the union of the usual complex upper half-plane with the set ℚ ∪ {∞}. The group 𝕊𝕃_{2}(ℤ) acts naturally on H^{∗} by linear fractional transformations; the quotient is the modular curve of level one, which (a little idiosyncratically, but for reasons of consistency) we will denote by X_{0}(1). It is well known that 𝕊𝕃_{2}(ℤ) acts transitively on ℚ∪{∞}, and we denote the corresponding point of X_{0}(1) simply by ∞, and refer to it as the cusp of X_{0}(1) (although of course it is not a singular point; the usage of this term is purely for historical reasons). Let us also write Y_{0}(1) = X_{0}(1) \ {∞}. It is well known that the points of Y_{0}(1) are in a one-to-one correspondence with the isomorphism classes of elliptic curves over ℂ; the orbit of a point τ in the upper half-plane corresponds to the elliptic curve E_{τ} := ℂ/Λ_{τ} , where Λ_{τ} is the lattice 2πiτℤ + 2πiℤ. (The 2πi factors are included just for the purposes of having a good normalization when it comes to comparing the analytic and algebraic theory of elliptic curves.)

From the algebraic theory of elliptic curves, we know that elliptic curves over any field are also classified up to isomorphism by their j-invariant. If we write j(τ) := j(E_{τ}), then (essentially tautologically, if one has a sufficient understanding of X_{0}(1) as a moduli space; see [9] for example) the function j(τ) is an analytic function on the open subset Y_{0}(1), which induces an analytic isomorphism Y_{0}(1) ⟶^{∼} 𝔸^{1}. The function j(τ) has a simple pole at ∞, and thus this isomorphism extends to an isomorphism

X_{0}(1) ⟶^{∼} ℙ^{1},

and by virtue of this isomorphism we will also regard j as a coordinate on X_{0}(1), and refer to X_{0}(1) as the j-line.

The curve X_{0}(2) is obtained by taking the quotient of H^{∗} by the congruence subgroup Γ_{0}(2) of 𝕊𝕃_{2}(ℤ). There are two orbits of Γ_{0}(2) on ℚ ∪ {∞}, the orbit of the point 0 and that of the point ∞. We refer to these points as the cusps of X_{0}(2), and we label them simply by 0 and ∞. We also write Y_{0}(2) = X_{0}(2) \ {0, ∞}.

Of course the points Y_{0}(2) also have an interpretation in terms of moduli of elliptic curves. Namely, given a point τ in the upper half-plane, we can form the curves E_{τ} and E_{2τ}, and observe that there is an isogeny between them:

(1.1)  E_{τ} ⟶ E_{2τ}

via the map z mod Λ_{τ} ↦ 2z mod Λ_{2τ} . The kernel of this isogeny has order two, and is generated by the point πi mod Λ_{τ}. One easily checks that if we modify τ by an element of Γ_{0}(2) then the isomorphism class of this two-isogeny remains unchanged, and that in fact the points of Y_{0}(2) are in bijection with the isomorphism classes of two-isogenies of elliptic curves over ℂ.

We were able to explicitly describe X_{0}(1) as an algebraic curve: it is the jline. Is there a similar description of X_{0}(2)? Here is a first attempt: since the orbit of a point τ under Γ_{0}(2) determines the two-isogeny (1.1) up to isomorphism, we certainly know its source and target up to isomorphism, and so to the orbit τ mod Γ_{0}(2) in Y_{0}(2) we can associate the two complex numbers j(τ) and j(2τ), which do classify the isomorphism class of E_{τ} and E_{2τ}. Thus we get a map

(1.2)  Y_{0}(2) ⟶ Y_{0}(1) × Y_{0}(1) ⟶^{∼} 𝔸^{2}.

Since the source of this map is one-dimensional over ℂ, its image in Y_{0}(1) × Y_{0}(1) must be a curve, whose equation will express a (somewhat complicated, as it turns out) relation between j(τ) and j(2τ). Such relations are classically called modular equations. From our point of view this modular equation has two disadvantages: firstly, it is a little hard to compute explicitly; secondly, the map (1.2) is not an isomorphism onto its image – it turns out that the image has singularities. What this means in modular terms is that for certain special choices of τ, there is one or more two-isogeny between E_{τ} and E_{2τ} which is not isomorphic to the isogeny (1.1).

p-ADIC GEOMETRY OF MODULAR CURVES 3

There is an alternative approach to describing X_{0}(2) as an algebraic curve, which we now present. We first proceed algebraically: thus we assume given a two isogeny ψ : E_{1} ⟶ E_{2} between two elliptic curves over ℂ. Let ω_{2} be a regular differential on E_{2}, and let ω_{1} := ψ^{∗}ω_{2} be the pulled-back regular differential on E_{1}.

Recall the algebraic definition of modular forms (of level one): a modular form f of weight k is a “rule” which to any pair consisting of an elliptic curve E and a non-zero regular differential ω on E attaches a number f(E, ω) depending only on the isomorphism class of the pair (E, ω), such that for any non-zero scalar λ, f(E, λω) = λ^{−k}f(E, ω), and which “behaves well in families”. (We won’t make precise the meaning of this statement here; see [10] or [25].)

There is a canonical modular form of weight 12, the discriminant Δ (see [10]). Returning to our two isogeny ψ : E_{1} ⟶ E_{2}, we define

j_{2}(ψ) = 2^{12} \frac{Δ(E_{1}, ω_{1})}{Δ(E_{2}, ω_{2})}.

Note that if we multiply ω_{2} by a non-zero scalar λ, both the numerator and denominator of j_{2}(ψ) are scaled by the same amount λ^{−12}, and so j_{2}(ψ) remains invariant. Thus it really does depend only on the isogeny ψ, and not on the auxiliary choice of ω_{2}.

Now let us interpret this in the analytic picture: on E_{τ} = ℂ/Λ_{τ} there is a canonical differential ω_{τ} obtained by reducing the differential dz on ℂ modulo Λ. (This reduction is possible because the differential dz is invariant under translation by elements of the lattice Λ_{τ}.) The function

Δ(τ) := Δ(E_{τ} , ω_{τ})

is a modular form (in the classical analytic sense explained in [42] for example) of weight twelve and level one – in fact it is the unique cuspform of this weight and level. Its q-expansion is given by the famous formula

Δ(τ) = q ∏_{n=1}^{∞}(1 − q^{n})^{24}.

(Recall that in the context of modular forms, q denotes the exponential q = e^{2πiτ} . The product formula for the q-expansion of Δ is proved in [42], for example.) If we let ψ_{τ} denote the isogeny (1.1), then we see that ψ_{τ}^{∗}ω_{2τ} = 2ω_{τ} , and so

(1.3)  j_{2}(ψ_{τ}) = 2^{12} \frac{Δ(E_{τ} , 2ω_{τ})}{Δ(E_{2τ} , ω_{2τ})} = \frac{Δ(E_{τ} , ω)}{Δ(E_{2τ} , ω_{2τ})}.

We define the function j_{2} on Y_{0}(2) via the formula

j_{2}(τ) = j_{2}(ψ_{τ}) =^{(1.3)} \frac{Δ(τ)}{Δ(2τ)} = \frac{q ∏_{n=1}^{∞}(1 − q^{n})^{24}}{q^{2} ∏_{n=1}^{∞}(1 − q^{2n})^{24}} = q^{−1} ∏_{n=1}^{∞}(1 + q^{n})^{−24}.

This is an analytic function Y_{0}(2) ⟶ 𝔸^{1}, and from its q-expansion (and the easily verified fact that q = e^{2πiτ} is a uniformizer in the neighbourhood of the point ∞ of X_{0}(2)) we see that is has a simple pole at the point ∞ of X_{0}(2). Thus j_{2} necessarily extends to an isomorphism X_{0}(2) ⟶^{∼} ℙ^{1}. Because of this, we will also refer to X_{0}(2) as the j_{2}-line.

Note that we conclude incidentally that any two-isogeny ψ is determined up to isomorphism by the invariant j_{2}(ψ). One could also establish this fact (with more difficulty) by pure algebra, rather than resorting to the consideration of modular curves, as we did above. However, one motivation for introducing the modular curves (or moduli spaces in general) is to simplify the proof of results such as this.

Now that we have our algebraic description of X_{0}(2) as the j_{2}-line, it will be interesting to return to and reinterpret slightly the map (1.2). The first coordinate of this map extends to a map X_{0}(2) ⟶ X_{0}(1), which is the natural map arising from the inclusion of Γ_{0}(2) in 𝕊𝕃_{2}(ℤ). We denote this map by B_{1}. The second coordinate of (1.2) also extends to a map X_{0}(2) ⟶ X_{0}(1), which we denote by B_{2}. There is also a natural automorphism of X_{0}(2), given by the construction of dual isogenies: if ψ : E_{1} ⟶ E_{2} is a two-isogeny, it has a dual two-isogeny ψ̌ : E_{2} ⟶ E_{1}. If ψ is the isogeny ψ_{τ} of (1.1) then ψ̌_{τ} is the isogeny E_{2τ} ⟶ E_{τ} arising from the inclusion of lattices Λ_{2τ} ⊂ Λ_{τ}. This is isomorphic to the two-isogeny ψ_{−1/2τ} . Thus we see that constructing dual isogenies yields an automorphism of Y_{0}(2) given by the formula

τ mod Γ_{0}(2) ↦ \frac{−1}{2τ} mod Γ_{0}(2).

This extends to an automorphism of X_{0}(2) which interchanges 0 and ∞. We denote this automorphism by w_{2}; it is an involution (has order two), and is often referred to as the Atkin-Lehner involution. The map B_{2} is easily described via w_{2}: since w_{2} interchanges the source and target of a two-isogeny, we see that B_{2} = B_{1} ∘ w_{2}.

How do we describe w_{2} in terms of the coordinate j_{2} on X_{0}(2)? This is an easy computation, using the modularity properties of Δ; one could make it either algebraically or analytically, but the latter is probably simpler:

j_{2}(−1/2τ) = \frac{Δ(−1/2τ)}{Δ(−1/τ)} = \frac{2^{12}τ^{12}Δ(2τ)}{τ^{12}Δ(τ)} = \frac{2^{12}}{j_{2}(τ)}.

Note that from this formula we also find the value of j_{2} at the cusp 0: j_{2}(0) = j_{2}(w_{2}(∞)) = 2^{12}/j_{2}(∞) = 0, since j_{2} has a pole at ∞.

Here is another question: how do we describe the function B_{1} in terms of the coordinates j and j_{2}?

Lemma 1.4. The map B_{1} : X_{0}(2) ⟶ X_{0}(1) is described by the equation

(1.5)  j ∘ B_{1} = (j_{2} + 256)^{3}/j_{2}^{2}.

It is ramified at the point 512 = j_{2}(i) over the point 1728 = j(i) with degree two, at the point −256 = j_{2}((−1 + \sqrt{−3})/2) over the point 0 = j((−1 + \sqrt{−3})/2) with degree three, and at the point 0 = j_{2}(0) over the point ∞ = j(∞) with degree two.

Proof. The map from the upper half-plane to Y_{0}(1) is ramified over two points: the point j = 1728, corresponding to the 𝕊𝕃_{2}(ℤ) orbit of z = i, with ramification degree two, and the point j = 0, corresponding to the 𝕊𝕃_{2}(ℤ) orbit of z = (−1 + \sqrt{−3})/2, with ramification degree three.

The map from the upper half-plane to Y_{0}(2) is ramified over one point, corresponding to the Γ_{0}(2) orbit of z = (1 + i)/2, with ramification degree two.

Now Γ_{0}(2) has index three in 𝕊𝕃_{2}(ℤ), and so the map B_{1} has degree three, and from the preceding two paragraphs we see that the resulting degree three map Y_{0}(2) ⟶ Y_{0}(1) is ramified at the Γ_{0}(2) orbit of z = i, with ramification degree two, and is totally ramified over the point j = 0.

Of the two cusps ∞ and 0 on X_{0}(2), the cusp ∞ is unramified over the cusp ∞ on X_{0}(1) with respect to B_{1} (we saw this implicitly above, when we remarked that q is a uniformizer in a neighbourhood of ∞ on X_{0}(2), since it is also a uniformizer in a neighbourhood of ∞ on X_{0}(1)), and hence the cusp 0 must be ramified of degree two over the cusp ∞ on X_{0}(1).

Now pull back j via B_{1} to a function on X_{0}(2). Since we know the zeroes and poles of j on X_{0}(1), as well as the ramification structure of B_{1} above each of these points, we see that j ∘ B_{1} has a pole of order one at ∞, a pole of order two at 0, a zero of order three at j_{2}((−1 + \sqrt{−3})/2), and no other zeroes or poles. Thus j ∘ B_{1} is a scalar multiple of (j_{2} − a)^{3}/j_{2}^{2}, where a = j_{2}((−1 + \sqrt{−3})/2). One computes that

(j_{2} − a)^{3}/j_{2}^{2} = 1/q − (24 + 3a) + ⋯,

while

j = 1/q + 744 + ⋯

(see [42]). Comparing these expressions we find that a = −256 and hence that

j ∘ B_{1} = (j_{2} + 256)^{3}/j_{2}^{2}.

From this equation we see that −256 = j_{2}((−1 + \sqrt{−3})/2) is the point lying over 0 = j((−1 + \sqrt{−3})/2), and also that

j ∘ B_{1} − 1728 = (j_{2} + 64)(j_{2} − 512)^{2}/j_{2}^{2},

and thus, from the above description of the ramification of B_{1}, that j_{2}(i) = 512 and j_{2}((1 + i)/2) = −64. We have now verified the asserted formula for the map B_{1} and also the claims about its ramification. □

The equation B_{2} = B_{1} ∘ w_{2} allows us to compute that

(1.6) j ∘ B_{2} = ((j_{2} + 256)^{3}/j_{2}^{2}) ∘ w_{2} = (2^{12}/j_{2} + 256)^{3}/(2^{12}/j_{2})^{2} = (j_{2} + 16)^{3}/j_{2}.

With sufficient enthusiasm, one can eliminate j_{2} from the two equations (1.5) and (1.6) and thus find the equation relating j ∘ B_{1} and j ∘ B_{2}; this will be the modular equation which describes the image of the map (1.2). Classically, one of the roles of higher level modular curves such as X_{0}(2) (or more precisely, the modular functions such as j_{2} that are defined on them) was to simplify the shape of the modular equations. In modern terms, we can see this as being related to the simplification that comes from replacing a singular curve by its normalization.

2. Canonical subgroups of elliptic curves over ℂ_{2}. The Lefschetz principle assures us that algebraic geometry is the same when studied over any algebraically closed field of characteristic zero. In this section we will take our ground field to be the completion of the algebraic closure of the field of 2-adic numbers ℚ_{2}. This field (which is algebraically closed as well as complete) is denoted ℂ_{2}; it provides a natural location for conducting 2-adic analysis, just as ℂ provides a natural location for performing classical analysis. In particular, ℂ_{2} is equipped with a nonarchimedean absolute value, which we denote by | |, and which we normalize by the condition that |2| = 1/2 (although the normalization won’t be important).

By the Lefschetz principle, all the observations about moduli of elliptic curves over ℂ which we made in the preceding section apply equally well to elliptic curves over ℂ_{2}. Thus we see that there is j-line whose non-infinite points classify isomorphism classes of elliptic curves over ℂ_{2}, as well as a j_{2}-line, whose points (other than ∞ and 0) classify isomorphism classes of two-isogenies between elliptic curves over ℂ_{2}. There is an involution w_{2} of the j_{2}-line given by passing to the dual isogeny, there are two maps B_{1} and B_{2} mapping the j_{2}-line to the j-line, and all the formulas for w_{2}, B_{1} and B_{2} that we proved in the preceding section continue to hold true.

Let us consider the formula (1.5) for B_{1}. We can rewrite this in the form

\frac{256}{j ∘ B_{1}} = \frac{256/j_{2}}{(1 + 256/j_{2})^{2}} .

Now if |j_{2}| > |256|, then we may expand the right-hand side of this equation in a power series, and so obtain the equation

\frac{256}{j ∘ B_{1}} = ∑_{n=1}^{∞}(−1)^{n−1}n (\frac{256}{j_{2}})^{n} .

Since the leading coefficient of this series is 1 (and so in particular is non-zero) we see by the implicit function theorem (which in this non-archimedean context is just a formal manipulation of power-series) that B_{1} establishes an isomorphism between the region |j_{2}| > |256| on the j_{2}-line and the region |j| > |256| on the j-line. Each of these regions is a disk centred at the point ∞ on the appropriate line; let us denote them by D_{2} and D_{1} respectively. Thus we see that B_{1} induces a 2-adic analytic isomorphism from D_{2} to D_{1}, and so has a 2-adic analytic inverse B_{1}^{−1} : D_{1} ⟶ D_{2}. So what?

Well, in modular terms, this means that if E is an elliptic curve over ℂ_{2} whose jinvariant satisfies the inequality |j(E)| > |256|, then there is a naturally determined two-isogeny whose source is E. Equivalently, thinking of a two-isogeny as being determined by its kernel, we see that there is a naturally determined subgroup of E of order two. Since E has three distinct subgroups of order two, it seems pretty remarkable that there is any way to distinguish one of them as being naturally determined! (Caveat: since the point B_{1}^{−1}(j(E)) in D_{2} only determines a twoisogeny up to isomorphism, the subgroup of order two is determined only up to the application of automorphisms of E. Now if E has no automorphism besides ±1, then since these both fix any subgroup of E, we see that the isomorphism class of the isogeny does determine a well-defined subgroup of order two of E. What if E has extra automorphisms? The point j = 0 (whose corresponding elliptic curve has six automorphisms) is not in the disk D_{1}, so it remains to consider the point j = 1728 (whose corresponding elliptic curve has four automorphisms). Now if j(E) = 1728, let α denote an automorphism of E of order four. Then α − 1 has degree two as an endomorphism of E, and so α fixes exactly one subgroup of order two of E, and interchanges the other two. Since B_{1}^{−1}(1728) lies in D_{2}, we see by lemma 1.4 and its proof that B_{1}^{−1}(1728) = −64, and since this is not a ramification point of B_{1}, the kernel of the corresponding two-isogeny must be fixed by α. Thus we see that the caveat is no caveat at all: even taking into account possible extra automorphisms of E, we see that the map B_{1}^{−1} does determine an order two subgroup of every E with j(E) lying in D_{1}.) This subgroup is called the canonical subgroup of E.

Now we consider the composite ϕ := B_{2} ∘ B_{1}^{−1} : D_{1} ⟶ X_{0}(1), which is called the Deligne-Tate map. The map B_{1}^{−1} is described by a formula of the form 256/B_{1}^{−1}(j) = ∑_{n=1}^{∞} a_{n}(256/j)^{n}, for certain integers a_{n} such that a_{1} = 1. The map B_{2} is described by (1.6), which can be rewritten in the form

\frac{256}{j} ∘ B_{2} = \frac{(16/j_{2})^{2}}{(1 + 16/j_{2})^{3}} .

Thus we find that

\frac{256}{ϕ(j)} = ∑_{n=2}^{∞} b_{n} (\frac{16}{j})^{2} ,

for some integers b_{n} such that b_{1} = 1. Actually, although ϕ is defined on all of D_{1}, this power-series formula only converges on the subdisk D′_{1} defined by the inequality |j_{2}| > |16|; it shows that the restriction of ϕ to D′_{1} is a degree two map whose image is D_{1} and which satisfies the formula

(2.1)  |ϕ(j)| = |j|^{2}

for any point j of D′_{1}.

What is the modular interpretation of the Deligne-Tate map? Well, if we start with an elliptic curve E such that j(E) lies in D_{1}, then B_{1}^{−1}(j) associates to E its canonical subgroup C. Then B_{2} associates to the pair (E, C) the target of the isogeny whose kernel is C, which is just E/C. Thus in modular terms ϕ associates to any elliptic curve E with j(E) in D_{1} its quotient by its canonical subgroup. Now if j = j(E) in fact lies in D′_{1}, we saw that ϕ(j) lies in D_{1}, and so E/C also has a canonical subgroup, which pulls back to a subgroup C′ of E of order four.

Lemma 2.2. In the above notation, C′ is a cyclic subgroup of E

Proof. Either C′ is a cyclic subgroup of E of order four, or it is the full two-torsion subgroup of E. In the latter case, we would see that E/C′ is isomorphic to E, and thus that E ⟶ E/C and E/C ⟶ E/C′ are dual isogenies.

On the other hand, both of these isogenies have j_{2} lying in D_{2} (since they are in the image of B_{1}^{−1}), and the first even satisfies |j_{2}| > |16|, because |j(E)| > |16|, and B_{1}^{−1} preserves absolute values. If |j_{2}| > |16|, then |2^{12}/j_{2}| < |256|, and so 2^{12}/j_{2} does not lie in D_{2}. Thus they cannot be dual isogenies, and we have proved the lemma. □

More generally, if we let D_{1}^{(n)} denote the disk in the j-line determined by the inequality |j| > |2^{2^{3−n}}|, then we see that ϕ^{n} : D_{1}^{(n)} ⟶ D_{1}, and that as a consequence if E is an elliptic curve over ℂ_{2} for which j(E) lies in D_{1}^{(n)}, then E has a canonical subgroup of order 2^{n+1}, which is cyclic (by the easily proved analogue of lemma 2.2).

3. The parameter q . This section and the next are a digression on some points of theory which we will need to better understand the preceding calculations. The subject of our first digression is an alternative approach to describing elliptic curves, using the parameter q rather than τ. This leads to a more theoretical interpretation of the q-expansion of a modular form, and to a modular interpretation of the cusp ∞ on the j-line.

Let us begin by considering the affine j-line Y_{0}(1) over the complex numbers. If τ is a point in H its image in Y_{0}(1) corresponds to the elliptic curve E_{τ} = ℂ/(2πiτℤ + 2πiℤ). We can take this quotient in two steps, by noting that the exponential function induces an isomorphism ℂ/2πiℤ ⟶^{∼} ℂ^{×} (where ℂ^{×} denotes the multiplicative group of non-zero complex numbers.) Under the exponential, the lattice 2πiτℤ + 2πiℤ has image equal to the cyclic subgroup q^{ℤ} of ℂ^{×}, where q = e^{2πiτ}. Thus we can describe E_{τ} as the quotient ℂ^{×}/q^{ℤ}. Thinking of ℂ^{×} as a cylinder of infinite length (which it is topologically), this is just the familiar description of a torus as being obtained by gluing together the ends of a finite length cylinder (where in our case, the finite length cylinder will be a fundamental domain for the action of q^{ℤ} on ℂ^{×}), as in the following picture:

We will use this description of elliptic curves to obtain a geometric description of the q-expansions of modular forms. We begin with a discussion of differentials. Let z denote the coordinate on ℂ and t the coordinate on ℂ^{×}, so that t = e^{z}. Then the differential dz on ℂ (which is invariant under the action of 2πiℤ by translation) descends to the differential dt/t on ℂ^{×}, and since this differential is invariant under the action of q^{ℤ} by multiplication, it descends to a differential on ℂ^{×}/q^{ℤ}.

Let D^{∗} denote the punctured open unit disk, consisting of those complex numbers q such that 1 > |q| > 0. As q ranges over all elements of D^{∗}, the elliptic curves ℂ^{×}/q^{ℤ} form a family of elliptic curves lying over D^{∗}, and the differential dt/t on ℂ^{×} descends to a relative differential on this family. Denote this family of elliptic curves by E_{T ate}, and this relative differential by ω_{T ate}. If f is any modular form of some weight k, then we may evaluate f on each member of the family, and thus obtain a function on D^{∗}. More succinctly, we evaluate f on the elliptic curve E_{T ate} over the base D^{∗} to obtain an element f(E_{T ate}, ω) of the ring of functions on D^{∗} (necessarily holomorphic – this is a consequence of the condition that f behave well in families). But such a function is just a convergent power-series in q, and this is precisely the q-expansion of the modular form f. Thus the existence of q-expansions of modular forms is related to the existence of the family of elliptic curve E_{T ate} over D^{∗}, equipped with its canonical relative differential ω_{T ate}.

We now turn to an analysis of the cusp ∞ in X_{0}(1). As τ tends towards the point ∞ in the boundary of H^{∗}, the value of q tends to zero. Thus in the preceding description of E_{τ} , the circumference of the finite cylinder is remaining fixed, while its length is tending towards zero. What happens when τ reach the point ∞? One way to describe what happens is to remember that the circumferential circle of this cylinder, and the circle obtained by gluing the endpoints of a longitudinal interval, represent two independent generators of the fundamental group of E_{τ}. What we just observed is that as τ goes to ∞, one of these loops stays a fixed length, while the length of the other tends to zero. This makes it reasonable to think of the point τ = ∞ as corresponding to the curve obtained by completely shrinking one of these loops to a point, as in the following picture:

In short, the point ∞ on X_{0}(1) corresponds to a rational curve with one node. In fact one can show that there is a canonical way to extend the family of elliptic curves E_{T ate} over the punctured disk D^{∗} to a family of curves over the (unpunctured) open unit disk D whose fibre at q = 0 is the above rational nodal curve.

If one is working over a field K different from ℂ, the above analytic construction of the family E_{T ate} is not available. However, one can construct in an analogous way an elliptic curve E_{T ate} over the field K((q)) (which one thinks of as the ring of functions on the formal punctured disk), equipped with a canonical nowhere-zero differential ω_{T ate}. Furthermore, the j-invariant of E_{T ate} is an element of K((q)) given precisely by the usual formula for j as a function of q [10, 25]. Letting 𝔾_{m} denote the multiplicative group over K((q)), one can even describe E_{T ate} as a quotient 𝔾_{m}/q^{ℤ}, in a suitable category of geometric objects, and (if t denotes the coordinate on 𝔾_{m}) ω_{T ate} is obtained by descending the differential dt/t on 𝔾_{m}. Finally, the elliptic curve E_{T ate} extends to a curve over the ring K[[q]], whose fibre over q = 0 is a nodal curve. (See [11] for the rather technical details of the construction of E_{T ate} over K[[q]].)

One can unify the treatment of elliptic curves and singular curves of the type that correspond to the cusp j = ∞ by introducing the notion of generalized elliptic curve [11]. Since such a uniform treatment is possible, at various points in the following text the conceptual distinction between elliptic curves and rational nodal curves will become a little blurred.

If f is any modular form over K then one defines the q-expansion of f to be the element of K((q)) obtained by evaluating f on the pair (E_{T ate}, ω_{T ate}) [10,25]. The q-expansion principle asserts that if f and g are two modular forms of the same weight having the same q-expansion, then they are equal. This is more or less a direct corollary of the fact that the map from the function field K(j) of X_{0}(1) to K((q)) obtained by taking the j-invariant of E_{T ate} is an injection. (See [25] for the details.) The usual requirement of the q-expansion of a modular form, that it lie in k[[q]] rather than just k((q)), in the light of this section corresponds to the fact that modular forms extend from functions on elliptic curves to functions on generalized elliptic curves (and so can be evaluated on the fibre of E_{T ate} over the point q = 0, for example) [11].

4. The Hasse invariant. In this section we recall briefly the theory of the Hasse invariant of elliptic curves in positive characteristic. We will need to treat elliptic curves defined over rings other than fields. Thus we let p be a prime and R a ring of characteristic p.

Let E be an elliptic curve over R. Recall that there is canonically associated to E another elliptic curve E^{(p)} and a canonical isogeny π : E ⟶ E^{(p)} of degree p, the Frobenius. Furthermore, if ω is a nowhere-zero regular differential on E, then it induces canonically a nowhere-zero regular differential ω^{(p)} on E^{(p)}.

The easiest way to explain these constructions is to note that given E and ω we can find (at least locally over R) a Weierstrass equation

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}

for E such that ω = dx/(2y + a_{1}x + a_{3}). Then E^{(p)} is given by the Weierstrass equation

y^{2} + a_{1}^{p}xy + a_{3}^{p}y = x^{3} + a_{2}^{p}x^{2} + a_{4}^{p}x + a_{6}^{p},

the differential ω^{(p)} is equal to dx/(2y + a_{1}^{p}x + a_{3}^{p}), and the isogeny π is the map (x, y) ↦ (x^{p}, y^{p}).

Let us suppose that E and ω are given as above, and let π̌ be the dual isogeny to π. Then we can construct two differentials on E^{(p)}: the nowhere-zero regular differential ω^{(p)}, and also the pulled-back differential (π̌)^{∗}ω. Define A(E, ω) to be that element of R such that

(π̌)^{∗}ω = A(E, ω)ω^{(p)}.

If we multiply ω by a unit λ of R, then we see that (λω)^{(p)} = λ^{p}ω^{(p)}, and consequently A(E, λω) = λ^{1−p}A(E, ω). Thus A is a modular form of weight p − 1 in characteristic p.

Now suppose that R is a field of characteristic p. In this case, although the value of A(E, ω) depends on the choice of both E and ω, the property of A(E, ω) being zero or non-zero depends only on E, independent of the choice of ω. If it is non-zero, we say that E is ordinary; if it is zero, we say that E is supersingular. It is relatively easy to determine the supersingular j-invariants modulo any prime p. There are only finitely many isomorphisms classes of such curves; in fact their j-invariants all lie in the finite field 𝔽_{p^{2}}. Furthermore, there is a precise formula for their number, as well as a precise formula for a polynomial of which their j-invariants are the solutions (see [24]).

The remainder of this section is devoted to calculating the q-expansion of the Hasse invariant. Recall from the preceding section that the Tate curve E_{T ate} in characteristic p is the quotient (in an appropriate sense) of the multiplicative group 𝔾_{m} over 𝔽_{p}((q)) by the cyclic subgroup q^{ℤ}. Since the multiplicative group is already defined over 𝔽_{p}, we see that in computing E_{T ate}^{(p)} we just have to raise q to the p^{th} power, and so E_{T ate}^{(p)} is isomorphic to the quotient of 𝔾_{m} by the cyclic subgroup q^{pℤ}, the Frobenius isogeny is given by t mod q^{ℤ} ↦ t mod q^{pℤ}, and its dual π̌ is just the natural map 𝔾_{m}/q^{pℤ} ⟶ 𝔾_{m}/q^{ℤ}. Recall that ω_{T ate} equals dt/t, so that ω_{T ate}^{(p)} is again dt/t, and also (π̌)^{∗}ω_{T ate} = dt/t. We conclude from this that A(E_{T ate}, dt/t) = 1. In other words, the q-expansion of A is just the constant 1!

Note that this would be quite impossible in characteristic zero: the only modular forms over ℂ with constant q-expansions are those of weight zero. On the other hand, it is the fact that the weight p − 1 modular form A and the weight zero modular form 1 have the same q-expansion which gives rise to the possibility of congruences of modular forms of different weights, and the beautiful theory of padic families of modular forms. (See the discussion of section 8 below and especially the references cited therein.)

5. Return to X_{0}(2). In this section we try to shed some theoretical light on the calculations of section 2, to pave the way for the discussion in the case of a general prime p.

Let O denote the ring of integers in ℂ_{2}, that is, the ring of elements r for which |r| ≤ |1|; geometrically, O is the closed unit disk of ℂ_{2}. The open unit disk of elements r such that |r| < |1| is the maximal ideal 𝔪 of the valuation ring O, and the quotient field k of O by 𝔪 is an algebraic closure of the field 𝔽_{2}, which we denote by k.

Reducing modulo 𝔪 extends to a map from the projective j-line over ℂ_{2} to the projective j-line over k, which we call the specialization map. This map has the following modular interpretation: if E is an elliptic curve such that j(E) lies in O then E has a model with coefficients in O with good reduction modulo 𝔪, and the reduction of j(E) modulo 𝔪 is just the j-invariant of the reduction of this model module 𝔪. On the other hand, if j(E) ∈ ℂ_{2} \ O then any model of E over O has singular reduction modulo 𝔪, and the reduction of j(E) modulo 𝔪 is the point ∞, which as we saw in section 3 corresponds to isomorphism class of a generalized elliptic curve which is singular. Finally, the point ∞ on the j-line corresponds to the singular generalized elliptic curve over ℂ_{2}, any model over O of which certainly has singular reduction modulo 𝔪, and the point ∞ certainly reduces to the point ∞.

Let us (as is customary) extend the phrase ordinary reduction to include either good ordinary reduction or bad reduction. (One should really require bad multiplicative reduction. However, ℂ_{2} is algebraically closed, and so any E with bad reduction has a model with multiplicative reduction.) Since j = 0 is the unique supersingular j-invariant modulo 2, we see that the closed disk |j| ≥ |1| is the set of j-invariants in X_{0}(1) having ordinary reduction. We denote this disk by X_{0}(1)^{h} (h is for Hasse). The complement of X_{0}(1)^{h} in X_{0}(1) is the open disk |j| < |1|, and consists of the set of j-invariants having good supersingular reduction. We call this the supersingular disk in X_{0}(1).

An examination of equations (1.5) and (1.6) shows that the preimage of X_{0}(1)^{h} under both B_{1} and B_{2} is equal to the union of the two disks |j_{2}| ≥ 1 and |j_{2}| ≤ |2^{12}|. We denote the former disk by X_{0}(2)_{∞}^{h} (because it contains the cusp ∞) and the latter by X_{0}(2)_{0}^{h} (because it contains the cusp 0); their union we denote simply by X_{0}(2)^{h}. Note that X_{0}(2)^{h} is preserved by w_{2}, and that w_{2} interchanges X_{0}(2)_{∞}^{h} and X_{0}(2)_{0}^{h}. The fact that X_{0}(2)^{h} is preserved by w_{2} (which is equivalent to the fact that X_{0}(1)^{h} has the same preimage under either B_{1} or B_{2}) reflects the fact that the property of being ordinary or supersingular is an invariant of an isogeny class of elliptic curves in characteristic two.

The complement of X_{0}(2)^{h} in X_{0}(2) is the open annulus |1| > |j_{2}| > |2^{12}|. Its points correspond to those two-isogenies whose source (or equivalently target) has good supersingular reduction. We refer to it as the supersingular annulus in X_{0}(2). picture:

The disk X_{0}(1)^{h} is contained in the disk D_{1} on which B_{1}^{−1} is defined, and we see that B_{1}^{−1} restricts to yield an isomorphism X_{0}(1)^{h} ⟶^{∼} X_{0}(2)_{∞}^{h}; in particular, every elliptic curve with ordinary reduction has a canonical subgroup.

We can characterize the disk X_{0}(1)^{h} as the set of j-invariants whose reduction modulo 𝔪 has non-zero Hasse invariant. The rest of this section is devoted to finding an analogous description of the larger disk D_{1}.

Let E_{4} denote the weight four Eisenstein series on 𝕊𝕃_{2}(ℤ), whose q-expansion is

E_{4} = 1 + 240 ∑_{n=1}^{∞} σ_{3}(n)q^{n},

where σ_{3}(n) = ∑_{d|n} d^{3}. Recall that this modular forms is actually well-defined in all

characteristics (see [10], where it is denoted c_{4}) and that the j-invariant satisfies (indeed is defined by) the equation j = E_{4}^{3}/Δ.

If E is an elliptic curve over ℂ_{2} with good reduction, and ω is a differential on E with non-zero reduction modulo 𝔪, then for any modular form f over ℂ_{2} we may compute f(E, ω), and the absolute value |f(E, ω)| is independent of the choice of ω (because any two such ω differ only by multiplication by a unit of O). Thus we are entitled to write simply |f(E)| in place of |f(E, ω)|. For example, Δ(E, ω) must be a unit in O (since it reduces to the discriminant of the reduction of E, which is non-zero), and so |Δ(E)| = |1|, yielding the formula

|j(E)| = \frac{|E_{4}^{3}(E)|}{|Δ(E)|} = |E_{4}^{3}(E)|.

Hence we see that |j(E)| > |256| if and only if |E_{4}(E)| > |2^{8/3}|.

We now wish to relate E_{4} to the Hasse invariant. The Hasse invariant A is a modular form of weight one defined modulo 2. Thus A^{4} is a modular form of weight one defined modulo 8, whose q-expansion is the constant 1. (The point being that if x is a number well-defined modulo 2, than x^{4} is well-defined modulo 8.) On the other hand, E_{4} reduces to a modular form modulo 8 whose q-expansion is also equal to the constant 1 (because 8 divides 240). Thus by the q-expansion principle, we see that E_{4} ≡ A^{4} mod 8. Thus |E_{4}(E)| > |2^{8/3}| if and only if |A(E)|^{4} > |2^{8/3}| (because |2^{8/3}| > |8|, and so the second equality can be checked after reducing modulo 8, where A^{4} makes sense) if and only if |A(E)| > |2^{2/3}| (because |2^{2/3}| > |2|, and so the second inequality can be checked after reducing modulo 2, where A makes sense). Here |A(E)| is denoting the valuation of the Hasse invariant of E reduced over O/2 (which is non-Noetherian highly non-reduced local ring!), not E reduced over k = O/𝔪.

Thus we see that the annulus |1| ≥ |j| > |256| consists precisely of those jinvariants for which the corresponding elliptic curve E satisfies |A(E)| > |2^{2/3}|. We may think of these as the j-invariants of elliptic curves with “not too supersingular” reduction modulo two (in the words of [25]; recall that the annulus |j| = |1| corresponds precisely to the elliptic curves having good ordinary reduction), and so the disk D_{1} of elliptic curves consists of those elliptic curves having not too supersingular reduction (in the preceding sense) along with the elliptic curves having bad reduction.

6. The theory for arbitrary primes p. In this section we explain the generalization to an arbitrary prime p of the results presented above in the case p = 2. Let ℂ_{p} denote the completion of the algebraic closure of ℚ_{p}, let O denote its ring of integers, and let 𝔪 denote the maximal ideal of O.

We let Y_{0}(p) denote the modular curve whose points parameterize p-isogenies between elliptic curves, and X_{0}(p) its completion. Over ℂ one can construct X_{0}(p) as the quotient H^{∗}/Γ_{0}(p), and so see that it has two cusps, 0 and ∞. As p increases the genus of X_{0}(p) also increases, and in particular X_{0}(p) is not a rational curve unless p = 13 or p < 11. Thus one cannot in general describe X_{0}(p) by a single parameter, as we described X_{0}(2) by the parameter j_{2}.

Nevertheless, Y_{0}(p) is an affine curve, and so it has an affine ring, which is generated by parameters which classify p-isogenies up to isomorphism (and which can be constructed explicitly via modular forms). By the Lefschetz principle, we see that we may equally regard Y_{0}(p) as a curve over ℂ_{p}, since the moduli problem of classifying p-isogenies will depend on the same invariants whether solved over ℂ or over ℂ_{p}. Then X_{0}(p) will be the completion of Y_{0}(p), a smooth projective curve over ℂ_{p}.

The process of passing from an isogeny to its dual yields an involution of X_{0}(p), which we denote by w_{p}. Passing to the j-invariant of the source of an isogeny is a morphism B_{1} : X_{0}(p) ⟶ X_{0}(1). We let B_{2} denote the composition B_{1} ∘ w_{p}.

We let X_{0}(1)^{h} denote the set of j-invariants in X_{0}(p) corresponding to elliptic curves with ordinary reduction (that is, either bad reduction or good ordinary reduction). The complement of X_{0}(1)^{h} in X_{0}(1) is a disjoint union of disks: each disk is a congruence class modulo 𝔪 of j-invariants which are congruent to a particular supersingular j-invariant in characteristic p. We refer to these disks as the supersingular disks in X_{0}(1).

We let X_{0}(p)^{h} denote the preimage of X_{0}(1)^{h} under B_{1}^{−1}. Just as in the case of p = 2, and for the same reason (that being ordinary or supersingular is an isogeny class invariant in characteristic p), X_{0}(p)^{h} is invariant under w_{p}. Furthermore, it is the union of two connected components which are interchanged by w_{p}, which we label X_{0}(p)_{∞}^{h} (the component containing ∞) and X_{0}(p)_{0}^{h} (the component containing 0), and the restriction of B_{1} to X_{0}(p)_{∞}^{h} is an isomorphism onto X_{0}(1)^{h}. (In order to interpret connected component in a more than intuitive sense, one must use the language of rigid analytic geometry.) The complement of X_{0}(p)_{∞}^{h} and X_{0}(p)_{0}^{h} in X_{0}(p) is a disjoint union of annuli, each the preimage under B_{1} of one of the supersingular disks in X_{0}(1). We refer to these as the supersingular annuli in X_{0}(p). There is the following picture that goes with this discussion, in which one thinks of X_{0}(p) as being obtained by gluing together two copies of the “sphere with holes” X_{0}(1)^{h} (in the guise of X_{0}(p)_{∞}^{h} and X_{0}(p)_{0}^{h}) via the supersingular annuli:

To prove all the facts just stated, one must study the modular curve X_{0}(p) as a scheme over ℤ_{p}, as in [11,30]. The assertions then follow in a fairly standard fashion from the known structure of X_{0}(p) over ℤ_{p}; see the discussion and references in [1].

For now, note that since B_{1} : X_{0}(p)_{∞}^{h} → X_{0}(1)^{h} is an isomorphism, it has an inverse B_{1}^{−1} : X_{0}(1)^{h} → X_{0}(p)^{h}, and thus any elliptic curve with ordinary reduction is equipped with a canonical subgroup of order p. Just as in the case of p = 2, there is a larger set D_{1} containing X_{0}(1)^{h} to which the map B_{1}^{−1} extends, and which we now describe.

Let A denote the Hasse invariant in characteristic p. If E is any elliptic curve with good reduction, we may consider its reduction modulo p (which is an elliptic curve over the ring O/p), and the inequality |A(E)| > |p^{p/(p+1)}| makes sense, since |p^{p/(p+1)}| > |p|. Define D_{1} to the the subset of X_{0}(1) consisting of the set of points in X_{0}(1)^{h} together with those j-invariants of elliptic curves having good reduction whose Hasse invariant satisfies this inequality. Then just as in the case of p = 2, the map B_{1}^{1} extends to a map B_{1}^{−1} : D_{1} → X_{0}(p) which is an isomorphism between D_{1} and its image D_{2}. If j(E) lies in D_{1} then the map B_{1}^{−1} endows E with a canonical subgroup of order p.

Let D_{1}^{(n)} denote the disk which is the union of X_{0}(1)^{h} and those j-invariants of good reduction whose corresponding elliptic curve satisfies |A(E)| > |p^{1/p^{n−1}(p+1)}|. Just as in the case p = 2, the Deligne-Tate map ϕ := B_{2} ∘ B_{1}^{−1} is a degree p map from D_{1}^{(n)} onto D_{1}^{(n−1)}, and in particular if j(E) lies in D_{1}^{(n)} then E has a canonical cyclic subgroup of order p^{n}.

When written in terms of the uniformizing parameter q at ∞ the Deligne-Tate map has the form q ↦ q^{p}. We will give a rather lengthy explanation of this fact, which will lead us toward an explanation of the existence of the Deligne-Tate map itself.

The Deligne-Tate map corresponds in modular terms to taking the quotient of an elliptic curve E by its canonical subgroup. Let us first try to understand why it is that the Deligne-Tate map is defined on the formal neighbourhood of the cusp ∞ on X_{0}(1). In other words, why does E_{T ate} have a canonical subgroup? Or again, why is there a map B_{1}^{−1} inverse to B_{1} defined on the formal neighbourhood of ∞?

This form of the question we can answer: the map B_{1} is a unramified at the point ∞ on X_{0}(p), and so induces an isomorphism of the formal neighbourhood of ∞ in X_{0}(p) with the formal neighbourhood of ∞ in X_{0}(1). The map B_{1}^{−1} is simply the inverse to this isomorphism. The existence of B_{1}^{−1} means that the Tate curve E_{T ate} = 𝔾_{m}/q^{ℤ} is equipped with a canonical subgroup of order p. To compute this subgroup, we will look at the analytic picture over ℂ.

If τ is a point in the upper half-plane then the image of τ in X_{0}(p) is the isogeny E_{τ} ⟶ E_{pτ} given by z mod Λ_{τ} ↦ pz mod Λ_{pτ} . The kernel of this isogeny is the

group \frac{2πi}{p}ℤ/ℤ. Exponentiating this description of the isogeny, we see that it can

also be described as the map ℂ^{×}/q^{ℤ} ⟶ ℂ^{×}/q^{pℤ} given by t ↦ t^{p}, whose kernel is the image of the group μ_{p} of p^{th} roots of unity in 𝔾_{m}/q^{ℤ}.

We conclude that the canonical subgroup of E_{T ate} = 𝔾_{m}/q^{ℤ} is the image of μ_{p} in 𝔾_{m}/q^{ℤ}. The quotient of 𝔾_{m}/q^{ℤ} by this group is isomorphic to 𝔾_{m}/q^{pℤ}, and thus is a Tate elliptic curve with parameter q^{p}. Hence in terms of the parameter q around the cusp ∞, the Deligne-Tate map is given by the formula q ↦ q^{p}.

The above argument actually provides the key insight as to the existence of canonical subgroups in general. Let us first explain this for ordinary elliptic curves. If E is an elliptic curve over ℂ_{p} with bad (multiplicative) reduction, then E can be obtained as a Tate curve by specializing the parameter q to a point of 𝔪 (this is Tate’s theory of uniformization of elliptic curves with multiplicative reduction), and the canonical subgroup of E = ℂ_{p}^{×}/q^{ℤ} will simply be the image of μ_{p}. If E is an elliptic curve over ℂ_{p} with good ordinary reduction, then E does not admit a description in terms of the Tate curve, but the situation is almost as good. Let E denote a model for with good reduction modulo 𝔪. Then one finds that the formal group of E (thought of as a formal group over O) is isomorphic to the formal multiplicative group 𝔾̂_{m}. (This is a consequence of the fact that the reduction modulo 𝔪 of each of these formal groups is of height one, so that they are isomorphic over k, together with a Newton’s method/Hensel’s lemma argument, which allows one to lift this isomorphism to an isomorphism over O.) Now the p-torsion in 𝔾̂_{m} is just the group μ_{p}, and so this isomorphism yields a copy of μ_{p} as a subgroup of E. This is the canonical subgroup of E.

To describe the canonical subgroup of a non-ordinary elliptic curve E whose jinvariant lies in D_{1} is more difficult. One examines the formal group of E (whose reduction modulo 𝔪 is now of height two) and uses the bound on the Hasse invariant of E (which one knows by virtue of the assumption that j(E) lies in D_{1}) to construct a canonically determined subgroup of this formal group of order p. The details can be found in [25].

7. p-adic modular forms. Recall again the algebraic definition of a modular form of weight k and of level one defined over the field ℂ_{p}: it is a rule that attaches to a pair (E, ω) consisting of an elliptic curve and a non-zero regular differential defined over ℂ_{p} a number f(E, ω) which satisfies the weight k transformation rule and which behaves well in families. From now on we will refer to such forms as classical modular forms.

A p-adic modular form of weight k is such a rule which is defined only on those pairs (E, ω) for which E has ordinary reduction (that is, as we said above, has either bad or good ordinary reduction modulo 𝔪). Thus any classical modular form also gives rise to a p-adic modular form, but there are also many p-adic modular forms which are not classical modular forms. For example, if p = 2 and E has ordinary reduction, then E_{4}(E, ω) ≠ 0, and so E_{4}(E, ω)^{−1} is well-defined. Thus E_{4}^{−1} is a 2-adic modular form of weight -4. Another way to construct p-adic modular forms for any prime p is to notice that ordinary elliptic curves have canonical subgroups of order p^{n} for all n, and so any classical modular form on Γ_{0}(p^{n}) gives rise to a p-adic modular form of level one.

The space of all p-adic modular forms of weight k (for some fixed p and k) is a p-adic Banach space, and just as in the classical theory this space has Hecke operators acting on it. Unfortunately, the action of these operators is rather hard to control, and one does not get, and cannot expect to get, a very good spectral theory. Thus there is not a very good theory of p-adic Hecke eigenforms.

To deal with this, one looks at a more refined type of p-adic modular form. An overconvergent p-adic modular form is a rule of the usual type which is defined on those pairs (E, ω) for which j(E) lies in D_{1}^{(n)} for some n. (The n is fixed for any particular overconvergent form, but may vary from form to form.) All our examples of p-adic modular forms given above are actually overconvergent modular forms: for example, we saw that E_{4} is never zero on the disc D_{1} in X_{0}(2), and so E_{4}^{−1} is defined on that disc. Also, any classical modular form on Γ_{0}(p^{n}) for some n yields an overconvergent modular form defined on D_{1}^{(n)}, since the elliptic curves with j-invariant in this region are precisely those which have a canonical subgroup of order p^{n}. Here is an example of a p-adic modular form which is not overconvergent: the “weight two Eisenstein series of level one” is the q-expansion E_{2} = 1 + 24 ∑_{n=1}^{∞} σ(n)q^{n}, where σ(n) = ∑_{d|n} d. This is not a classical modular form at all (it doesn’t transform correctly under the substitution τ ↦ −1/τ), but is the q-expansion of a p-adic modular form for every p (see [25], in which it is denoted by P), which is not overconvergent for any prime p [6].

One can again define Hecke operators acting on overconvergent modular forms, and one finds that something special happens. If one works with the Hecke operator U_{p} (which on q-expansions is defined by U_{p}(∑_{n} a_{n}q^{n}) = ∑_{n} a_{np}q^{n})) rather than T_{p}, then there is a good theory of Hecke eigenforms, because the operator U_{p} is a completely continuous (or, in alternative language, compact) operator on the space of overconvergent modular forms, and so has a good spectral theory. More precisely, for any non-zero λ ∈ ℂ_{p}, there is a finite-dimensional space of overconvergent modular forms on which U_{p} has eigenvalue λ. This space will be preserved by the Hecke operators T_{ℓ} (as ℓ ranges over all primes different from p), and to construct Hecke eigenforms with U_{p}-eigenvalue λ it suffices to diagonalize these commuting operators on this finite-dimensional vector space.

The reason that U_{p} is completely continuous is the following: recall that the Deligne-Tate map is given in terms of the parameter q by the formula ϕ(q) = q^{p}. Thus one sees that the operator U_{p} can be described geometrically via the trace of the Deligne-Tate map. More precisely, it is 1/p times this trace. It is simplest to explain this for U_{p} acting on modular forms of weight zero, which is to say modular functions. Then an overconvergent modular function is simply an analytic function f on one of the regions D_{1}^{(n)}. We claim that

U_{p}(f)(j) = \frac{1}{p} ∑_{ϕ(j′)=j} f(j′)

for any point j ∈ D_{1}^{(n)}. To see this, it suffices to verify the formula for those j lying in the residue disc about ∞, where we can compute in terms of q; it will follow in general by analytic continuation. The claim then follows from the above formula for the action of U_{p} on q-expansions, together with the following simple piece of algebra:

\frac{1}{p} ∑_{q^{′p}=q} ∑_{n} a_{n}q^{′n} = ∑_{n} a_{np}q^{n}.

(See [25] for a precise form of this argument.) Why does this formula imply that U_{p} is completely continuous? Well, note that if j ∈ D_{1}^{(n)} and ϕ(j′) = j, then j′ ∈ D_{1}^{(n+1)}. Thus computing U_{p}(f) involves restricting f from the region D_{1}^{(n)} to the proper subregion D_{1}^{(n)}, and such restriction operators are always completely continuous. (This is Montel’s theorem from complex analysis being applied in the (simpler) p-adic setting.)

The preceding discussion shows that the fact that the Deligne-Tate map is defined on the regions D_{1}^{(n)} which extend some way into the supersingular annuli is fundamental to obtaining a good spectral theory for overconvergent p-adic modular forms. This explains the importance of the Deligne-Tate map in the theory of p-adic modular forms, and hence why we have devoted our efforts in this appendix to describing the ideas behind the construction of this map.

8. Guide to the literature. This final section contains a brief review of the literature on the p-adic theory of modular curves and modular forms, which I hope will be helpful to someone trying to enter the field. I have tried to present as accurate an account of the development and current state of the field as I can, within the limits of my own understanding of these matters. I apologize in advance for any omissions or oversights that I may inadvertently have made.

The p-adic aspects of the theory of modular curves that we have discussed in this appendix seem to make their first appearance in the literature in [12], in which Dwork constructs and studies “by hand” the Deligne-Tate map on the region D_{1}. He states that the existence of this map was conjectured by Tate (based on a calculation for p = 2; I don’t know if it was a similar calculation to that of section 2 above) and first proved in general by Deligne. In subsequent articles [13, 14] Dwork studied the spectral theory of the completely continuous operator U_{p} on the space of overconvergent modular forms of level zero. One of the points emphasized in [14] in particular is the importance of restricting ones attention to overconvergent forms if one hopes to obtain a reasonable spectral theory.

In an independent line of research, Swinnerton-Dyer [48] began the systematization of the theory of congruences of q-expansions of modular forms (a subject which seems to have begun with the work of Ramanujan), by studying the ring of modular forms modulo p. This work is also reported on in [43]. In [44] Serre extends these results to develop a theory of congruences of modular forms modulo arbitrary powers of p, and introduces the notion of p-adic modular forms.

In [25] Katz gives a systematic presentation of the p-adic geometry of the modular curves, including the construction of canonical subgroups of “not too supersingular elliptic curves” (which construction he attributes to Lubin) and the consequent construction of B_{1}^{−1} and the Deligne-Tate map, as well as of the theory of congruences of modular forms modulo p. The results of this article, and the later article [26] (which extends the techniques of [25] to define generalized p-adic modular functions, a notion that includes as a special case the p-adic modular forms of [25] and [44]), subsumed most of those of Dwork, Serre and Swinnerton-Dyer mentioned above, and also generalized them to modular forms of arbitrary level.

However, although [25] in some sense unified the various existing p-adic theories, the applications of these theories were in two different directions. On the one hand the papers [27, 28, 29, 44] used the theory of congruences of modular forms to construct p-adic analytic families of modular forms (essentially families of Eisenstein series) and hence to construct p-adic L-functions (which appeared as the special values of these Eisenstein series, either at the cusp ∞ or at certain special j-values corresponding to complex-multiplication elliptic curves). On the other hand, the papers [48, 49] were concerned with using the same theory of congruences not to construct families of modular forms, but rather to understand the image of the Galois representations attached to Hecke eigenforms (as constructed in [9]). Thus their main concern was not congruences between arbitrary modular forms, but the possible congruences that could arise between Hecke eigenforms. This was something that was not dealt with in the theory of [26], and this may go some way to explaining the remark on this paper made in the introduction of [49].

Studying properties of Galois representations via congruences of modular forms turned out to be a very fruitful idea. One key direction of research was the investigation of congruences between cuspidal Hecke eigenforms and Eisenstein series. This is the main theme of the seminal paper [34], and formed the basis for a successful attack on the so-called main conjecture of Iwasawa theory over the field ℚ, beginning in [41], continuing in [50], and culminating with the proof of the main conjecture in [39]. The investigation of possible congruences among the cuspidal eigenforms themselves proved to be an equally important topic, a discussion of which would unfortunately take us too far afield, and which is provided by Ribet’s article in this volume.

In Hida’s papers [22,23] the properly p-adic aspect of the theory of congruences made its resurgence. In these pivotal papers Hida connected the study of congruences of eigenforms and the associated Galois representations with the study of p-adic families of modular forms by constructing p-adic analytic families of padic Hecke eigenforms and attaching to them p-adic analytic families of Galois representations. (See also [40] for a discussion of Hida’s construction and a more refined analysis of these Galois representations.) The main technical constraint on Hida’s results is that they only construct families of ordinary eigenforms; that is, eigenforms whose U_{p} eigenvalues are p-adic units.

The influence of Hida’s theory was enormous. It allowed a simplification of the proof of the main conjecture of Iwasawa theory, by rephrasing it as the question of analyzing the intersection locus of the Eisenstein family with the cuspidal part of Hida’s family. By extending Hida’s theory to the context of Hilbert modular forms, Wiles was able to prove the main conjecture for arbitrary totally real fields [51]; this harks back to Serre’s study of p-adic L-functions as the constant term of families of Eisenstein series. Other developments included the construction of families of p-adic L-functions attached to Hida families of cuspforms [20, 33, 35]. Greenberg and Stevens [20] used these L-functions to prove the weight two case of the conjecture of Mazur, Tate and Teitelbaum [38]. (See also [21], which presents the main ideas of their argument in a simplified setting.) of p-adic modular forms, Hida’s work also motivated Mazur to develop his theory of deformations of Galois representations [36] (for more on which, see the main body of this article!), which proved decisive for the further development of the theory of p-adic modular forms.

In [18] Gouvea used Mazur’s theory in order to associate a p-adic Galois representation to each p-adic modular form, whether ordinary or not. This work marked the beginning of line of research aimed at constructing p-adic analytic families of p-adic Hecke eigenforms and Galois representations, analogous to those constructed by Hida, but in the non-ordinary situation. That such families should exist was the principal conjecture of [19]. This work also raised a number of questions related to overconvergent p-adic modular forms, refocusing attention on an aspect of the p-adic theory that had languished since the appearance of papers of Dwork cited above. The p-adic analytic viewpoint favoured by Dwork reemerged in a series of papers by Coleman [1, 2, 3, 4]. The first of these papers relates to the conjecture of [38], while the second and third show that an overconvergent p-adic eigenform whose U_{p}-eigenvalue is not too divisible by p (in a sense depending on the weight of the form in question) is necessarily classical (extending the analogous result in the ordinary case, due to Hida [23], and hence answering one of the questions of [18]).

The current course of research in the theory of p-adic modular forms has been set by the papers [4] and [7]. In the first of these papers, Coleman analyzes the variation of the spectral theory of the U_{p} operator on overconvergent p-adic modular forms as a function of the weight and hence constructs p-adic analytic families of Hecke eigenforms, in particular proving in a qualitative form the conjectures of [19]. In [7] this construction is globalized to construct for each prime p a p-adic rigid analytic curve called the eigencurve. This is, roughly speaking, the universal parameter space for p-adic analytic families of overconvergent Hecke eigenforms of finite slope (that is, whose U_{p}-eigenvalue is non-zero; the slope of a Hecke eigenform is the p-adic valuation of its U_{p}-eigenvalue). The deformation theory of Galois representations plays a key role in the construction of the eigencurve, and the universal family of finite-slope eigenforms comes equipped with a p-adic analytic family of p-adic Galois representations.

With the construction of the eigencurve, it appears that the various directions of research that instigated the development of the theory of p-adic modular forms have achieved synthesis. The overconvergent forms are understood as being the necessary “glue” that binds classical Hecke eigenforms together into analytic families. Furthermore, the existence of positive slope families will certainly have many number-theoretic applications. One such example is [46], in which Stevens develops a theory of p-adic L-functions in the context of Coleman’s families, and uses them [47] to prove the higher-weight case of the conjecture of [38] (in the more precise form stated in [1]).

Yet the construction of the eigencurve raises as many questions as it answers. For example, while the structure of Hida’s families of ordinary forms is very precisely understood, the structure of the higher-slope parts of the eigencurve is essentially a mystery (the references [8, 15, 45] provide some information on a small part of this structure for certain small values of p). Relatedly, the quantitative aspects of the conjectures of [19] remain unresolved; all that is known are some very special cases dealt with in [8, 15, 45].

Another question is that of the existence of p-adic analytic families of eigenforms of infinite slope. Such forms, whose U_{p}-eigenvalue vanishes, aren’t accessible via the standard spectral theory techniques available for the study of the completely continuous operator U_{p}. One natural question that arises in this context is: are there infinite slope forms that can be written as the limit of finite slope forms? Such forms would correspond to “punctures” in the eigencurve, which would be filled in by the limiting infinite slope form. Coleman [5] has shown that forms obtained by twisting finite slope forms with the Teichmüller character are of this type. In the same reference, Coleman reports on a computation of Stein suggesting that some non-twist infinite slope forms might also be of this type. As to whether infinite slope forms move in p-adic families, some computations for forms on Γ_{0}(4)

[16] suggest that the answer could be yes, but essentially nothing is known in this direction.

One of the most intriguing and general questions raised by the construction of the eigencurve is that of the intrinsic Galois-theoretic interpretation of the Galois representations that it parameterizes (these are just the Galois representations attached to overconvergent p-adic Hecke eigenforms, and this problem was already posed in [18]). In [7] it is shown that the eigencurve is the rigid analytic Zariski closure (in an appropriate ambient space, which is essentially a deformation space of Galois representations) of those points corresponding to classical modular forms. The Galois representations associated to such forms are conjecturally characterized as being those which are potentially semistable (see [17] and the references contained therein for a discussion of this conjecture and the notion of potential semistability). However no analogous intrinsic characterization of the non-classical points on the eigencurve is known, even on the conjectural level (see [31, 32] for some investigations in this direction). Finding such a description of the eigencurve, which would not depend on the “extrinsic” construction via the theory of overconvergent p-adic modular forms, is related to the problem of generalizing the construction of the eigencurve to other settings. For a discussion of what such settings might be, and the possible relations to Iwasawa theory and a general theory of p-adic L-functions, see [37].

References

1. R. F. Coleman, A p-adic Shimura isomorphism and p-adic periods of modular forms, Contemp. Math. 165 (1994), 21-51.

2. R. F. Coleman, Classical and overconvergent modular forms, Invent. Math 124 (1996), 215-241.

3. R. F. Coleman, Classical and overconvergent modular forms of higher level, Journal de Théorie des Nombres de Bordeaux 7 (1995), 151-163.

4. R. F. Coleman, p-adic Banach spaces and families of modular forms, Invent. Math. 127 (1997), 417-479.

5. R. F. Coleman, Conjectures and rumours of conjectures, notes to some talks, posted at

http://www.math.berkeley.edu/~coleman/Sems/Sp00/coleman.html (2000).

6. R. F. Coleman, F. Q. Gouvea, and N. Jochnowitz, E_{2}, Θ, amd overconvergence, IMRN No. 1 (1995), 23-41.

7. R. F. Coleman and B. Mazur, The eigencurve, Galois representations in arithmetic algebraic geometry, London Math. Soc. Lecture Note Series, vol. 254, 1998, pp. 1-113.

8. R. F. Coleman, G. Stevens and J. Teitelbaum, Numerical experiments on families of p-adic modular forms, preprint.

9. P. Deligne, Formes modulaires et représentations ℓ-adiques, SLN 179 (1971), 139-172.

10. P. Deligne, Courbes elliptiques: formulaire d’après J. Tate, SLN 476 (1975), 53-73.

11. P. Deligne and M. Rapoport, SLN 349 (1973), 143-316.

12. B. Dwork, p-adic cycles, Pub. Math. IHES 37 (1969), 27-115.

13. B. Dwork, On Hecke polynomials, Invent. Math. 12 (1971), 249-256.

14. B. Dwork, The U_{p} operator of Atkin on modular functions of level 2 with growth conditions, SLN 350 (1973), 57- 67.

15. M. Emerton, 2-adic modular forms of minimal slope, Harvard Ph. D. thesis (1998).

16. M. Emerton, unpublished computations (1998).

17. J. -M. Fontaine and B. Mazur, Geometric Galois representations, Elliptic curves, modular forms, and Fermat’s last theorem (Hong Kong 1993), 1995, pp. 41-78.

18. F. Q. Gouvea, Arithmetic of p-adic modular forms, SLN 1304 (1988).

19. F. Q. Gouvea and B. Mazur, Families of modular eigenforms, Mathematics of Computation 58 (1992), 793-805.

20. R. Greenberg and G. Stevens, p-adic L-functions and p-adic periods of modular forms, Invent. Math. 111 (1993), 401-447.

21. R. Greenberg and G. Stevens, On the conjecture of Mazur, Tate and Teitelbaum, Contemp. Math. 165 (1994), 183-211.

22. H. Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. Ec. Norm. Sup. (4) 19 (1986), 231-273.

23. H. Hida, Galois representations into 𝔾𝕃_{2}(ℤ_{p}[[X]]) attached to ordinary cusp forms, Inv. Math. 85 (1986), 545-613.

24. J. -I. Igusa, Class number of a definite quaternion with prime discriminant, Proc. Nat. Acad. Sci. USA 44 (1958), 312-314.

25. N. Katz, p-adic properties of modular schemes and modular forms, Lecture Notes in Math., vol 350, Springer-Verlag, Berlin and New York, 1973, pp. 69-190.

26. N. Katz, Higher congruences between modular forms, Ann. of Math. 101 (1975), 332-367.

27. N. Katz, p-adic L functions via moduli, Proceedings of Symposia in Pure Mathematics, vol. 29, American Math Society, Providence, Rhode Island, 1975, pp. 479-506.

28. N. Katz, p-adic interpolation of real analytic Eisenstein series, Ann. of Math. 104 (1976), 459-571.

29. N. Katz, The Eisenstein measure and p-adic interpolation, Am. J. Math. 99 (1977), 238-311.

30. N. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Annals of Math. Studies, vol. 108, 1985.

31. M. Kisin, Periods for p-adic modular forms, preprint (1999).

32. M. Kisin, p-adic modular forms and the Fontaine-Mazur conjecture, preprint (2000).

33. K. Kitagawa, On standard p-adic L-functions of families of elliptic cusp forms, Contemp. Math. 165 (1994), 81-110.

34. B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math., Inst. Hautes Etud. Sci. 47 (1977), 33-186.

35. B. Mazur, Two variable p-adic L-functions, unpublished manuscript (1985).

36. B. Mazur, Deforming Galois representations, Galois groups over ℚ, MSRI Publ., vol. 16, 1999.

37. B. Mazur, The theme of p-adic variation, Frontiers and perspectives in mathematics, 2000.

38. B. Mazur, J. Tate and J. Teitelbaum, On p-adic analogs of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), 1-48.

39. B. Mazur and A. Wiles, Class fields of abelian extensions of ℚ, Inv. Math. 76 (1984), 179-330.

40. B. Mazur and A. Wiles, On p-adic analytic families of Galois representations, Comp. Math. 59 (1986), 231-264.

41. K. A. Ribet, A modular construction of unramified p-extensions of ℚ(μ_{p}), Invent. Math. 34 (1976), 151-162.

42. J. -P. Serre, A course in arithmetic, Springer, 1973.

43. J. -P. Serre, Congruences et formes modulaires [après H. P. F. Swinnerton-Dyer], SLN 317 (1973), 319-338.

44. J. -P. Serre, Formes modulaires et fonctions zêta p-adiques, Lecture Notes in Math., vol 350, Springer-Verlag, Berlin and New York, 1973, pp. 191-268.

45. L. Smithline, Berkeley Ph. D. thesis (2000).

46. G. Stevens, Rigid analytic modular symbols, unpublished manuscript (1994).

47. G. Stevens, Coleman’s L-invariant and families of modular forms, unpublished manuscript (1996).

48. H. P. F. Swynnerton-Dyer, On ℓ-adic representations and congruences for coefficients of modular forms, SLN 350 (1973), 1-55.

49. H. P. F. Swynnerton-Dyer, On ℓ-adic representations and congruences for coefficients of modular forms, SLN 601 (1977), 63-90.

50. A. Wiles, Modular curves and the class group of ℚ(ζ_{p}), Invent. Math. 58 (1980), 1-35.

51. A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. 131 (1990), 493-540.

department of mathematics, university of michigan, ann arbor

   E-mail address: emerton@math.lsa.umich.edu