Let π be a cuspidal automorphic representation of Sp_2n over Q which is holomorphic discrete series at infinity, and χ a Dirichlet character. Then one can attach to π an orthogonal p-adic Galois representation ρ of dimension 2n+1. Assume ρ is...

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Joint IAS/PU Number Theory

In the early 2000’s Ruijsenaars and Felder-Varchenko have introduced the elliptic gamma function, a remarkable multivariable meromorphic q-series that comes from mathematical physics. It satisfies modular functional equations under the group SL3(Z)...

We ask the question, “how does the infinite q-Pochhammer symbol transform under modular transformations?” and connect the answer to that question to the Stark conjectures. The infinite q-Pochhammer symbol transforms by a generalized factor of...

This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams.

Based on random matrix theory, Conrey-Farmer-Keating-Rubinstein-Snaith have conjectured precise asymptotics for moments of families of quadratic L...

The Fontaine-Mazur conjecture (proved by Kisin and Emerton) says that (under certain technical hypotheses) a Galois representation ρ:GalQ→GL2(Q⎯⎯⎯⎯p)

is modular if it is unramified outside finitely many places and de Rham at p. I will talk about...

Symmetric power functoriality is one of the basic cases of Langlands' functoriality conjectures and is the route to the proof of the Sato-Tate conjecture (concerning the distribution of the modulo p point counts of an elliptic curve over Q, as the...

The quantum unique ergodicity conjecture of Rudnick and Sarnak concerns the mass equidistribution in the large eigenvalue limit of Laplacian eigenfunctions on negatively curved manifolds. This conjecture has been resolved by Lindenstrauss when this...

In a recent machine learning based study, He, Lee, Oliver, and Pozdnyakov observed a striking oscillating pattern in the average value of the P-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland...

We prove this bound by first using the unitary Ichino-Ikeda formula of N. Harris to relate the central L-value to an automorphic period integral. There is a `trivial' bound for this integral, which turns out to correspond to the convexity bound for...

I discuss the spectral and arithmetic side of the relative trace formula of Kuznetsov type for congruence subgroups of SL(n, Z) with applications to automorphic density theorems. A particular focus is on properties of general Kloosterman sums as...