Joint IAS/PU Number Theory

Adjoint Selmer groups for polarized automorphic Galois representations

Patrick Allen
University of Illinois, Urbana-Champaign
October 15, 2015
Given the $p$-adic Galois representation associated to a regular algebraic polarized cuspidal automorphic representation, one naturally obtains a pure weight zero representation called its adjoint representation. Because it has weight zero, a conjecture of Bloch and Kato says that the only de Rham extension of the trivial representation by this adjoint representation is the split extension. We will discuss a proof of this case of their conjecture, under an assumption on the residual representation.

Motivic cohomology actions and the geometry of eigenvarieties

David Hansen
Columbia University
October 1, 2015
Venkatesh has recently proposed a fascinating conjecture relating motivic cohomology with automorphic forms and the cohomology of arithmetic groups. I'll describe this conjecture, and discuss its connections with the local geometry of eigenvarieties and nonabelian analogues of the Leopoldt conjecture. This is joint work with Jack Thorne.

On the formal degrees of square-integrable representations of odd special orthogonal and metaplectic groups

Atsushi Ichino
Kyoto University
February 5, 2015
The formal degree conjecture relates the formal degree of an irreducible square-integrable representation of a reductive group over a local field to the special value of the adjoint gamma-factor of its L-parameter. We prove the formal degree conjecture for odd special orthogonal and metaplectic groups in the generic case, which combined with Arthur's work on the local Langlands correspondence implies the conjecture in full generality. This is joint work with Erez Lapid and Zhengyu Mao.

Endoscopy theory for symplectic and orthogonal similitude groups

Bin Xu
Member, School of Mathematics
January 29, 2015
The endoscopy theory provides a large class of examples of Langlands functoriality, and it also plays an important role in the classification of automorphic forms. The central part of this theory are some conjectural identities of Harish-Chandra characters between a reductive group and its endoscopic groups. Such identities are established in the real case by Shelstad, but they are still largely unknown in the p-adic case due to our limited knowledge of characters in this case.

Level raising mod 2 and arbitrary 2-Selmer ranks

Chao Li
Harvard University
December 4, 2014
We prove a level raising mod $p = 2$ theorem for elliptic curves over $\mathbb Q$, generalizing theorems of Ribet and Diamond-Taylor. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families. We will begin by explaining our motivation from W. Zhang's approach to the $p$-part of the BSD conjecture. Explicit examples will be given to illustrate different phenomena compared to odd $p$. This is joint work with Bao V. Le Hung.

Weyl-type hybrid subconvexity bounds for twisted L-functions and Heegner points on shrinking sets

Matthew Young
Texas A & M University; von Neumann Fellow, School of Mathematics
November 20, 2014
One of the major themes of the analytic theory of automorphic forms is the connection between equidistribution and subconvexity. An early example of this is the famous result of Duke showing the equidistribution of Heegner points on the modular surface, a problem that boils down to the subconvexity problem for the quadratic twists of Hecke-Maass L-functions. It is interesting to understand if the Heegner points also equidistribute on finer scales, a question that leads one to seek strong bounds on a large collection of central values.

An algebro-geometric theory of vector-valued modular forms of half-integral weight

Luca Candelori
Lousiana State University
October 23, 2014
We give a geometric theory of vector-valued modular forms attached to Weil representations of rank 1 lattices. More specifically, we construct vector bundles over the moduli stack of elliptic curves, whose sections over the complex numbers correspond to vector-valued modular forms attached to rank 1 lattices. The key idea is to construct vector bundles of Schrodinger representations and line bundles of half-forms over appropriate `metaplectic stacks' and then show that their tensor products descend to the moduli stack of elliptic curves.

The standard \(L\)-function for \(G_2\): a "new way"

Nadya Gurevich
Ben-Gurion University of the Negev
October 2, 2014
We consider a Rankin-Selberg integral representation of a cuspidal (not necessarily generic) representation of the exceptional group \(G_2\). Although the integral unfolds with a non-unique model, it turns out to be Eulerian and represents the standard \(L\)-function of degree 7. We discuss a general approach to the integrals with non-unique models. The integral can be used to describe the representations of \(G_2\) for which the (twisted) \(L\)-function has a pole as functorial lifts. This is a joint work with Avner Segal.