NTS Fall 2012/Abstracts

September 13

Nigel Boston (UW–Madison)

Title: Non-abelian Cohen–Lenstra heuristics

Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian p-group A (p odd) arises as the p-class group of an imaginary quadratic field K is apparently proportional to 1/|Aut(A)|. The group A is isomorphic to the Galois group of the maximal unramified abelian p-extension of K. In work with Michael Bush and Farshid Hajir, I generalized this to non-abelian unramified p-extensions of imaginary quadratic fields. I shall recall all the above and describe a further generalization to non-abelian unramified p-extensions of H-extensions of Q, for any p, H, where p does not divide the order of H.

September 20

Simon Marshall (Northwestern)

Title: Multiplicities of automorphic forms on GL2

Abstract: I will discuss some ideas related to the theory of p-adically completed cohomology developed by Frank Calegari and Matthew Emerton. If F is a number field which is not totally real, I will use these ideas to prove a strong upper bound for the dimension of the space of cohomological automorphic forms on GL2 over F which have fixed level and growing weight.

September 27

Jordan Ellenberg (UW–Madison)

Title: Topology of Hurwitz spaces and Cohen-Lenstra conjectures over function fields

Abstract: We will discuss recent progress, joint with Akshay Venkatesh and Craig Westerland, towards the Cohen–Lenstra conjecture over the function field Fq(t). There are two key novelties, one topological and one arithmetic. The first is a homotopy-theoretic description of the "moduli space of G-covers with infinitely many branch points." The second is a description of the stable components of Hurwitz space over Fq, as a module for Gal(Fq/Fq). At least half the talk will be devoted to explaining why these objects are relevant to a very down-to-earth question like Cohen–Lenstra. If time permits, I'll explain what this has to do with the conjectures Nigel spoke about two weeks ago, and a bit about what Daniel is up to.

October 4

Sean Rostami (Madison)

Title: Centers of Hecke algebras

Abstract: The classification and construction of smooth representations of algebraic groups (over non-archimedean local fields) depends heavily on certain function algebras called Hecke algebras. The centers of such algebras are particularly important for classification theorems, and also turn out to be the home of some trace functions that appear in the Hasse–Weil zeta function of a Shimura variety. The Bernstein isomorphism is an explicit identification of the center of an Iwahori–Hecke algebra. I talk about all these things, and outline a satisfying direct proof of the Bernstein isomorphism (the theorem is old, the proof is new).

October 11

Tonghai Yang (Madison)

Title: Quaternions and Kudla's matching principle

Abstract: In this talk, I will explain some interesting identities among average representation numbers by definite quaternions and degree of Hecke operators on Shimura curves (thus indefinite quaternions).

October 18

Rachel Davis (Madison)

Title: On the images of metabelian Galois representations associated to elliptic curves

Abstract: For ℓ-adic Galois representations associated to elliptic curves, there are theorems concerning when the images are surjective. For example, Serre proved that for a fixed non-CM elliptic curve E/Q, for all but finitely many primes ℓ, the ℓ-adic Galois representation is surjective. Grothendieck and others have developed a theory of Galois representations to an automorphism group of a free pro-ℓ group. In this case, there is less known about the size of the images. The goal of this research is to understand Galois representations to automorphism groups of non-abelian groups more tangibly. Let E be a semistable elliptic curve over Q of negative discriminant with good supersingular reduction at 2. Associated to E, there is a Galois representation to a subgroup of the automorphism group of a metabelian group. I conjecture this representation is surjective and give evidence for this conjecture. Then, I compute some conjugacy invariants for the images of the Frobenius elements. This will give rise to new arithmetic information analogous to the traces of Frobenius for the ℓ-adic representation.

October 25

Shamgar Gurevich (Madison)

Title: Bounds on characters of SL(2, q) via Theta correspondence

Abstract: I will report on part from a joint project with Roger Howe (Yale). We develop a method to obtain effective bounds on the irreducible characters of SL(2, q). Our method uses explicit realization of all the irreducible representations via the Theta correspondence applied to the dual pair (SL(2, q), O), where O is an orthogonal group. If you want to learn what are all the notions in my abstract you are welcome to attend the talk. I will not assume any familiarity with the subject.

November 1

Lei Zhang (Boston College)

Title: Tensor product L-functions of classical groups of Hermitian type: quasi-split case

Abstract: We study the tensor product L-functions for quasi-split classical groups of Hermitian type times general linear group. When the irreducible automorphic representation of the general linear group is cuspidal and the classical group is an orthogonal group, the tensor product L-functions have been studied by Ginzburg, Piatetski-Shapiro and Rallis in 1997. In this talk, we will show that the global integrals are eulerian and finish the explicit calculation of unramified local L-factors in general.

November 8

Robert Harron (UW–Madison)

Title: Computing Hida families

Abstract: I will report on a joint project with Rob Pollack and four people you know well: Evan Dummit, Marton Hablicsek, Lalit Jain, and Daniel Ross. Our goal is to explicitly compute Hida families using overconvergent modular symbols. This grew out of a project at the Arizona Winter School and the basic idea is to study p-adic families of overconvergent modular symbols. I will go over the basic definitions and results starting from classical modular symbols and explain how one goes about encoding these objects on a computer. Aside from being able to compute formal q-expansions of Hida families, we can also compute the structure of the ordinary p-adic Hecke algebra, L-invariants, two-variable p-adic L-functions, etc. Several examples will be provided. The code is implemented in Sage.

November 15

Robert Lemke Oliver (Emory)

Title: Multiplicative functions with small sums

Abstract: Analytic number theory is in need of new ideas: for the very problem which motivated its existence – the distribution of primes – we have been unable to make progress in more than fifty years. Granville and Soundararajan have recently put forward a possible substitute for the seemingly intractable, though admittedly rich, theory of zeros of L-functions. They dub this new framework the pretentious view of analytic number theory, where the main objects of consideration are generic multiplicative functions, and the goal is to obtain deep theorems about the structure of the partial sums of such functions. In this talk, we consider multiplicative functions whose partial sums exhibit extreme cancellation. We will present two different lines of work about this problem. First, we develop what might be considered the pretentious framework to answer this question – notions of pretentious which permit the detection of power cancellation – which is joint work with Junehyuk Jung of Princeton University. Second, we consider a natural class of functions defined via the arithmetic of number fields, and we classify the members of this class which exhibit extreme cancellation; the proof of this is not at all pretentious.

November 29

Xin Shen (Minnesota)

Title: Unramified computation for automorphic tensor L-function

Abstract: In 1967 Langlands introduced the automorphic L-functions and conjectured their analytic properties, including the meromorphic continuation to C with finitely many poles and a standard functional equation. One of the important cases is the tensor L-functions for G × GLk where G is a classical group. In this seminar I will survey some approaches to this case via integral representations. I will also talk about my recent work on the unramified computation for L-functions of Sp2n × GLk for the non-generic case.

December 6

Sheng-Chi Liu (Texas A&M)

Title: Subconvexity and equidistribution of Heegner points in the level aspect

Abstract: We will discuss the equidistribution property of Heegner points of level q and discriminant D, as q and D go to infinity. We will establish a hybrid subconvexity bound for certain Rankin–Selberg L-functions which are related to the equdistribution of Heegner points. This is joint work with Riad Masri and Matt Young.

Organizer contact information

Robert Harron

Zev Klagsbrun

Sean Rostami

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