## University of Chicago Number Theory Seminar## Spring 2017: Tuesday 1:30-2:50pm, Eckhart 206 |

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Click here to see the schedule of previous quarters: Spring 2013 / Fall 2013 / Spring 2014 / Fall 2014 / Winter 2015 / Spring 2015 / Fall 2015 / Winter 2016 / Spring 2016

/ Spring 2016 / Fall 2016 / Winter 2017-
Date

Speaker

Title

March 28

**Preston Wake**

(*UCLA*)**Pseudorepresentations and the Eisenstein ideal**

In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. He proved a great deal about these congruences, but also posed a number of questions: how big is the space of cusp forms that are congruent to the Eisenstein series? How big is the extension generated by their coefficients? In joint work with Carl Wang Erickson, we give an answer to these questions using the deformation theory of Galois pseudorepresentations. The answer is intimately related to the algebraic number theoretic interactions between the primes N and p, and is given in terms of cup products (and Massey products) in Galois cohomology. ( Hide Abstract)April 4

**Evangelia Gazaki**

(*Michigan*)**A Tate duality theorem for local Galois symbols**

Let K be a p-adic field and M be a finite continuous Gal(Kbar/K)-module. Local Tate duality is a perfect duality between the Galois cohomology of M and the Galois cohomology of its dual module. In the special case when M is the module of the m-torsion points of an abelian variety A over K, Tate has a finer result.In this case the group H^{1}(K,M) has a "significant subgroup", namely there is a map A(K)/m &rarr H^1(K,M) induced by the Kummer sequence on A. Tate showed that under the perfect pairing for H^{1}, the orthogonal complement of A(K)/m is the corresponding part, A^{*}(K)/m, where A^{*}is the dual abelian variety of A. In this talk I will present an analogue of this classical result for H^{2}. The "significant subgroup" in this case will be given by a Galois symbol map, similar to the classical Galois symbol of the motivic Bloch-Kato conjecture, while the orthogonal complement under the Tate duality pairing will be given by an object of p-adic Hodge theory. (Hide Abstract)April 11

**Nicolas Templier***(Cornell)***Mirror symmetry for minuscule flag varieties**

We prove cases of Rietsch mirror conjecture that the Dubrovin-Givental quantum connection for projective homogeneous varieties is isomorphic to the pushforward D-module attached to Berenstein-Kazhdan geometric crystals. The idea is to recognize the quantum connection as Galois and the geometric crystal as automorphic. The isomorphism then comes from global rigidity results where a Hecke eigenform is determined by its local ramification. We reveal relations with the works of Gross, Frenkel-Gross, Heinloth-Ngo-Yun and Zhu on Kloosterman sheaves. The talk will keep the algebraic geometry prerequisite knowledge to a minimum by introducing the above concepts of "mirror" and "crystal" with the examples of CP^{1}, projective spaces and quadrics. Work with Thomas Lam. (Hide Abstract)April 18

**Ali Altuğ**

(*MIT*)**On the geometric side of the trace formula for GL(N)**

The Arthur-Selberg trace formula is one of the most fundamental and powerful tools in number theory and automorphic forms. For the general linear group GL(N), it gives a distributional identity I_{spec}(f)=I_{geom}(f) between a spectral and a geometric expansion of distributions on suitable functions f. The part that contribute discretely to the spectral side, I_{disc}, is central to the trace formula, and at the heart of I_{disc} is the so-called cuspidal part I_{cusp}(f). In many applications one is naturally lead to study I_{cusp} (or quantities related to I_{cusp}) and hence it is fundamental to get an "explicit" (and, in a certain sense, geometric) expression for the difference I_{geom}(f)-(I_{disc}(f)-I_{cusp}(f)). In this talk I will talk about the problem of isolating the contribution of I_{disc}(f)-I_{cusp}(f) in I_{geom}(f). I will present a solution of this in the case GL(2) and talk about recent work of Arthur for G=GL(N). I will also say a few words about what kind of obstacles one encounters when one tries to execute the GL(2) strategy in higher rank, and possible directions one can pursue to overcome these. (Hide Abstract)April 25

**Florian Sprung**

(*Princeton*)**What do special values of L-functions know? An approach to this question via sharp/flat objects.**

This talk has two components. 1) The first part is concerned with elliptic curves. If we knew the Iwasawa Main Conjecture for an elliptic curve at *any* prime p, then one consequence would be the full Birch and Swinnerton-Dyer conjecture in the case for analytic rank zero or one. The main conjecture is known at good ordinary primes. We give an overview of our sharp/flat objects and the proof of the Iwasawa Main Conjecture at good non-ordinary primes. 2) The second part is about modular forms. The Bloch-Kato-Shafarevich-Tate group for modular forms is an analogue of the class group for number fields. Like the class group, it is hard to compute the size of this group. In the 1950's, Iwasawa gave formulas for the size of the class group in an infinite tower of number fields. In this talk, we establish analogous formulas for modular forms, which contain Iwasawa's formula, but also two other terms, one of which is completely new. This is based on joint work with Rei Otsuki, involving sharp/flat p-adic L-functions. If time permits, we will draw a picture about a surprising phenomenon about these formulas. (There is some overlap with our talk at Northwestern the previous day, which will focus mostly on the Bloch-Kato-Shafarevich-Tate group.) (Hide Abstract)May 2

**No seminar**

May 9

**No seminar**

May 16

**Kęstutis Česnavičius**

*(Berkeley)***The Manin constant in the semistable case**

For an optimal modular parametrization J_0(n) --->> E of an elliptic curve E over Q of conductor n, Manin conjectured the agreement of two natural Z-lattices in the Q-vector space H^0(E, Omega^1). Multiple authors generalized his conjecture to higher dimensional newform quotients. I will discuss the semistable cases of the Manin conjecture and of its generalizations using a technique that establishes general relations between the integral p-adic etale and de Rham cohomologies of abelian varieties over p-adic fields. (Hide Abstract)May 23

**Brian Smithling**

*(Johns Hopkins)***An arithmetic intersection conjecture**

The arithmetic Gan-Gross-Prasad conjecture (AGGP) relates the non-vanishing of the special value of the derivative of a certain L-function to the non-vanishing of a linear functional on a Chow group of a Shimura variety. AGGP is itself based on conjectures of Beilinson and Bloch which seem to be out reach at the present; accordingly, there are no cases of AGGP known beyond dimension one. I will report on a variant of the hermitian version of AGGP, following previous work of Wei Zhang, which should be more accessible. This is joint work with M. Rapoport and W. Zhang. (Hide Abstract)May 30

**Vaidehee Thatte**

*(Queen's University)***Ramification theory for degree p extensions of arbitrary valuation rings in mixed characteristic (0,p)**

In classical ramification theory, we consider extensions of complete discrete valuation rings with perfect residue fields. We would like to study arbitrary valuation rings with possibly imperfect residue fields and possibly non-discrete valuations of rank \geq 1, since many interesting complications arise for such rings. In particular, defect may occur (i.e. we can have a non-trivial extension, such that there is no extension of the residue field or the value group). In this talk, we will discuss generalization and refinement of results in ramification theory when the residue characteristic p is positive, with focus on mixed characteristic (0,p). (Hide Abstract)

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