The successive minima of an order in a degree n number field are n real numbers encoding information about the Euclidean structure of the order. How many orders in degree n number fields are there with almost prescribed successive minima, fixed...

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Joint IAS/PU Number Theory

The behavior of quadratic twists of modular L-functions is at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves. Here we describe a proof of an asymptotic for the second moment of...

Given a set of integers, we wish to know how many primes there are in the set. Modern tools allow us to obtain an asymptotic for the number of primes, or at least a lower bound of the expected order, assuming certain strength Type-I information...

Consider the family of automorphic representations on some unitary group with fixed (possibly non-tempered) cohomological representation π0 at infinity and level dividing some finite upper bound. We compute statistics of this family as the level...

Tate reformulated the theory of the Riemann zeta function and its functional equation as the Mellin shadow of the Fourier transform on a certain space of function on the adeles. Conjecturally, Langlands' general automorphic L-functions and their...

Cohen, Lenstra, and Martinet gave conjectural distributions for the class group of a random number field. Since the class group is the Galois group of the maximum abelian unramified extension, a natural generalization would be to give a conjecture...

The Cohen-Lenstra heuristics give predictions for the distribution of the class groups of a random quadratic number field. Cohen and Martinet generalized them to predict the distribution of the class groups of random extensions of a fixed base field...

A Diophantine upper bound on the dimensions of certain spaces of holonomic functions was the main ingredient in our proof with Calegari and Tang of the 'unbounded denominators conjecture' (presented by Tang in last year's number theory seminar) from...

This lecture will partly survey branching laws for real and p-adic groups which often is related to period integrals of automorphic representations, discuss some of the more recent developments, focusing attention on homological aspects and the...

Half-integral weight modular forms are classical objects with many important arithmetic applications. In terms of automorphic representations, these correspond to objects on the metaplectic double cover of SL(2). In this talk, I will outline a...