In the 1940s Mahler initiated the program of determining the bass note spectrum Spec(P):={infx⎯⎯∈Λ∖0⎯⎯∣∣P(x⎯⎯)∣∣,Λ⊂ℝk a unimodular lattice} for some homogeneous form P. Understanding this spectrum is central in the geometry of numbers and offers a...

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Joint IAS/PU Number Theory

In the early 2000's, Bertolini and Darmon introduced a new technique to bound Selmer groups of elliptic curves via level raising congruences. This was the first example of what is now termed a "bipartite Euler system", and over the last decade we...

The second and fourth moments of the Riemann zeta function have been known for about a century, but the sixth moment remains elusive.

The sixth moment of zeta can be thought of as the second moment of a GL_3 Eisenstein series, and it is natural to...

It is known that a p-adic family of modular forms does not necessarily specialize into a classical modular form at weight one, unlike the modular forms of weight 2 or higher. We will explain how this obstruction to classicality leads to a "derived"...

We will describe emerging understanding of the structures related to the arithmetic of Zeta and Multizeta values for function fields through various results and conjectures.

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...

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...