# NTS ABSTRACTSpring2021

## Jan 28

 Monica Nevins Interpreting the local character expansion of p-adic SL(2) The Harish-Chandra—Howe local character expansion expresses the character of an admissible representation of a p-adic group G as a linear combination with complex coefficients of the (finitely many!) Fourier transforms of nilpotent orbital integrals $$\widehat{\mu}_{\mathcal{O}}$$ --- near the identity. Approaching from another direction: we can restrict the representation to any compact open subgroup K of G, obtaining its branching rules, which also describe the representation near the identity, in a different sense. We show that for G=SL(2,k), k a nonarchimedean local field, where the branching rules to maximal compact open subgroups K are known, each of these terms $$\widehat{\mu}_{\mathcal{O}}$$ can be interpreted as the character $$\tau_{\mathcal{O}}$$ of a representation of K, up to an error term arising from the zero orbit. Moreover, the irreducible components of $$\tau_{\mathcal{O}}$$ are explicitly constructed from the K -orbits in $$\mathcal{O}$$. This work in progress offers a conjectural alternative interpretation of branching rules of admissible representations.

## Feb 4

 Ke Chen On CM points away from the Torelli locus Coleman conjectured in 1980's that when g is an integer sufficiently large, the open Torelli locus T_g in the Siegel modular variety A_g should contain at most finitely many CM points, namely Jacobians of general curves of high genus should not admit complex multiplication. We show that certain CM points do not lie in T_g if they parametrize abelian varieties isogeneous to products of simple CM abelian varieties of low dimension. The proof relies on known results on Faltings height and Sato-Tate equidistributions. This is a joint work with Kang Zuo and Xin Lv.

## Feb 11

 Dmitry Gourevitch Relations between Fourier coefficients of automorphic forms, with applications to vanishing and to Eulerianity In recent works with H. P. A. Gustafsson, A. Kleinschmidt, D. Persson, and S. Sahi, we found a way to express any automorphic form through its Fourier coefficients, using adelic integrals, period integrals and discrete summation – generalizing the Piatetski-Shapiro – Shalika decomposition for GL(n). I will explain the general idea behind our formulas, and illustrate it on examples. I will also show applications to vanishing and Eulerianity of Fourier coefficients.

## Feb 18

 Eyal Kaplan The generalized doubling method, multiplicity one and the application to global functoriality One of the fundamental difficulties in the Langlands program is to handle the non-generic case. The doubling method, developed by Piatetski-Shapiro and Rallis in the 80s, pioneered the study of L-functions for cuspidal non-generic automorphic representations of classical groups. Recently, this method has been generalized in several aspects with interesting applications. In this talk I will survey the different components of the generalized doubling method, describe the fundamental multiplicity one result obtained recently in a joint work with Aizenbud and Gourevitch, and outline the application to global functoriality. Parts of the talk are also based on a collaboration with Cai, Friedberg and Ginzburg.