Number Theory
Organizers: Richard Taylor, Brian Conrad, Kannan Soundararajan, Xinwen Zhu, and Sarah Peluse.
Upcoming Events
Hecke operators play a fundamental role in understanding the arithmetic properties of modular and automorphic forms. Since the advent of the original Eichler-Shimura relation, it has been clear that the mod-p behavior of Hecke correspondences is crucial for such applications. However, one could…
A longstanding question in the theory of Shimura varieties concerns their perfectoidness at infinite level—a property that would reveal deep connections between étale and coherent cohomology. In this talk, we establish a criterion for perfectoidness via Sen theory, building on a new development…
Past Events
Eigenvarieties are parameter spaces of certain p-adic automorphic forms of varying weights. Part of the p-adic Langlands program aims to relate eigenvarieties to spaces of trianguline Galois representations. For definite unitary groups, this connection has been…
Abstract: If P is some real polynomial and L is some unimodular lattice, what is the infimum that the absolute value of P achieves on the non-trivial vectors of L? The set of these infima, when L ranges over all unimodular lattices, is called the bass note spectrum of P. For…
Classical Fourier theory describes measures on a locally compact abelian group in terms of functions on its Pontryagin dual. In this talk, I will explain an analogous theory for p-divisible rigid analytic groups (in the sense of Fargues) that recovers the Amice transform when applied to the open…
Random multiplicative functions are objects studied by both number theorists and probabilists in recent years. They are probabilistic models that are motivated by arithmetic problems in number theory. I will give an introduction and update on this rapidly developing area. Part of the talk is…
For general reductive groups over a p-adic number field, Fargues and Scholze constructed a (semi-simplified) local Langlands with many good properties. On the other hand, classical local Langlands correspondences are known for classical groups via endoscopy theory and theta lifting. We review…
Choose a finite group G and a number field F. We show that, given any large family of G-extensions of F, almost all are subject to a strong effective form of the Chebotarev density theorem. As one consequence, given a prime p, we are able to give nontrivial upper bounds for the size of the p-…
Abstract: We discuss two recent results which involve in their critical cases unexpected applications of nilsequences to certain point counting questions. The first gives an improved upper bound for the difference between consecutive squarefree numbers. The second concerns representations…
Abstract: I will describe recent work joint with Olga Balkanova and Dmitry Frolenkov on a restricted divisor function and its associated divisor problem. This problem shares properties with the usual Dirichlet divisor problem and the Hardy-Littlewood problem to count lattice points…
Abstract: Suppose that n is 0 or 4 mod 6. We show that there are infinitely many primes of the form p^2 + nq^2 with both p and q prime, and obtain an asymptotic for their number. In particular, when n = 4 we verify the `Gaussian primes conjecture' of Friedlander and Iwaniec.
Joint w. Ben…
Given an algebraic curve defined over a number field, one can define the Néron-Tate height on the Jacobian and prove its positivity. This height pairing and its positivity play important roles in the proof of the Mordell-Weil theorem, in Vojta's proof of the Mordell conjecture, and in the…