Joint Berkeley-Stanford Algebraic Number Theory

Elkies proved that an elliptic curve over the field of rational numbers has infinitely many supersingular reductions. The generalization of the 0-dimensional supersingular locus of the modular curve is the basic locus of a Shimura curve at a good prime.  In this talk, we generalize…

Abstract: In this talk, by using the trace formula method, I will prove a multiplicity formula of K-types for all representations of real reductive groups in terms of the Harish-Chandra character. 

Abstract: Let $(\lambda(n))_{n\geq 1}$ denote the sequence of coefficients of an automorphic $L$-function. Let $q$ be square free and $K:Z/qZ\mapsto C$ be a $q$-periodic function build from « trace functions «  from $\ell$-adic sheaves on affine lines in characteristics dividing $…

Abstract:

We will discuss a graph that encodes the divisibility properties of integers by primes. We show that this graph is shown to have a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity…

The mod p cohomology of locally symmetric spaces for definite unitary groups at infinite level is expected to realize the mod p local Langlands correspondence for GL_n. In particular, one expects the (component at p) of the associated Galois representation to be determined by cohomology as a…

Let S be a Shimura variety. The Andre-Oort conjecture posits that the Zariski closure of special points must be a sub Shimura subvariety of S. The Andre-Oort conjecture for A_g (the moduli space of principally polarized Abelian varieties) — and therefore its sub Shimura varieties — was proved by…

Let K be a p-adic field with absolute Galois group G_K. In 1980’s, Fontaine discovered the notion of being crystalline for finite Q_p-representations of G_K. This captures the property of having good reduction, analogously to being unramified in the l-adic case. The notion of crystalline…

The well-known classical Eichler-Shimura relation for modular curves asserts that the Hecke operator $T_p$ is equal, as an algebraic correspondence over the special fiber, to the sum of Frobenius and Verschiebung. Blasius and Rogawski proposed a generalization of this result for Shimura…

I will discuss a recent paper of mine, the aim of which is to count the number of prime solutions to Q(p_1,..,p_8) = N, for a fixed quadratic form Q and varying N. The traditional approach to problems of this type, the Hardy-Littlewood circle method, does not quite suffice. The main new idea is…