Joint IAS/PU Number Theory

Congruences between motives and congruences between values of $L$-functions

Olivier Fouquet
Université Paris-Sud
April 13, 2017
If two motives are congruent, is it the case that the special values of their respective $L$-functions are congruent? More precisely, can the formula predicting special values of motivic $L$-functions be interpolated in $p$-adic families of motives? I will explain how the formalism of the Weight-Monodromy filtration for $p$-adic families of Galois representations sheds light on this question (and suggests a perhaps surprising answer).

Basic loci of Shimura varieties

Xuhua He
University of Maryland; von Neumann Fellow, School of Mathematics
April 6, 2017

In mod-$p$ reductions of modular curves, there is a finite set of supersingular points and its open complement corresponding to ordinary elliptic curves. In the study of mod-$p$ reductions of more general Shimura varieties, there is a "Newton stratification" decomposing the reduction into finitely many locally closed subsets, of which exactly one is closed. This closed set is called the basic locus; it recovers the supersingular locus in the classical case of modular curves.

Galois Representations for the general symplectic group

Arno Kret
University of Amsterdam
March 30, 2017

In a recent preprint with Sug Woo Shin ( I construct Galois representations corresponding for cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. In this talk I will explain some parts of this construction that involve the eigenvariety.

On small sums of roots of unity

Philipp Habegger
University of Basel
March 9, 2017

Let $k$ be a fixed positive integer. Myerson (and others) asked how small the modulus of a non-zero sum of $k$ roots of unity can be. If the roots of unity have order dividing $N$, then an elementary argument shows that the modulus decreases at most exponentially in $N$ (for fixed $k$). Moreover it is known that the decay is at worst polynomial if $k < 5$. But no general sub-exponential bound is known if $k \geq 5$.

Arithmetic and geometry of Picard modular surfaces

Dinakar Ramakrishnan
California Institute of Technology; Visitor, School of Mathematics
December 8, 2016
Of interest are (i) the conjecture of Bombieri (and Lang) that for any smooth projective surface $X$ of general type over a number field $k$, the set $X(k)$, of $k$-rational points is not Zariski dense, and (ii) the conjecture of Lang that $X(k)$, is even finite if in addition $X$ is hyperbolic, i.e., there is no non-constant holomorphic map from the complex line $C$ into $X(C)$. We can verify them for the Picard modular surfaces $X$ which are smooth toroidal compactifications of congruence quotients $Y$ of the unit ball in $\mathbb C^2$.

Nonabelian Cohen-Lenstra heuristics and function field theorems

Melanie Wood
University of Wisconsin–Madison
November 17, 2016
The Cohen-Lenstra Heuristics conjecturally give the distribution of class groups of imaginary quadratic fields. Since, by class field theory, the class group is the Galois group of the maximal unramified abelian extension, we can consider the Galois group of the maximal unramified extension as a non-abelian generalization of the class group. We will explain non-abelian analogs of the Cohen-Lenstra heuristics due to Boston, Bush, and Hajir and work, some joint with Boston, proving cases of the non-abelian conjectures in the function field analog.

The Hasse-Weil zeta functions of the intersection cohomology of minimally compactified orthogonal Shimura varieties

Yihang Zhu
Harvard University
October 20, 2016
Initiated by Langlands, the problem of computing the Hasse-Weil zeta functions of Shimura varieties in terms of automorphic L-functions has received continual study. We will discuss how recent progress in various aspects of the field has allowed the extension of the project to some Shimura varieties not treated before.

Rational curves on elliptic surfaces

Douglas Ulmer
Georgia Institute of Technology
May 5, 2016
Given a non-isotrivial elliptic curve $E$ over $K = \mathbb F_q(t)$, there is always a finite extension $L$ of $K$ which is itself a rational function field such that $E(L)$ has large rank. The situation is completely different over complex function fields: For "most" $E$ over $K = \mathbb C(t)$, the rank $E(L)$ is zero for any rational function field $L = \mathbb C(u)$. The yoga that suggests this theorem leads to other remarkable statements about rational curves on surfaces generalizing a conjecture of Lang.

Standard conjecture of Künneth type with torsion coefficients

Junecue Suh
University of California, Santa Cruz
April 21, 2016
A. Venkatesh asked us the question, in the context of torsion automorphic forms: Does the Standard Conjecture (of Grothendieck's) of Künneth type hold with mod p coefficients? We first review the geometric and number-theoretic contexts in which this question becomes interesting, and provide answers: No in general (even for Shimura varieties) but yes in special cases.