Joint IAS/PU Number Theory

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.

Low-lying, fundamental, reciprocal geodesics

Alex Kontorovich
Rutgers University; Member, School of Mathematics
March 24, 2016
Markoff numbers give rise to extremely low-lying reciprocal geodesics on the modular surface, but it is unknown whether infinitely many of these are fundamental, that is, the corresponding binary quadratic form has fundamental discriminant. In joint work with Jean Bourgain, we unconditionally produce infinitely many low-lying (though not "extremely" so), reciprocal geodesics on the modular surface, settling a question of Einsiedler-Lindenstrauss-Michel-Venkatesh.

Iwasawa theory for the symmetric square of a modular form

David Loeffler
University of Warwick
March 10, 2016
Iwasawa theory is a powerful technique for understanding the link between the special values of L-functions and arithmetic objects (such as class groups of number fields, or Mordell-Weil groups of elliptic curves). In this talk I'll discuss what Iwasawa theory predicts for the symmetric square L-function attached to a modular form; and I'll describe some recent results (joint with Sarah Zerbes) confirming some of these conjectures, using the method of Euler systems.

Vanishing cycles and bilinear forms

Will Sawin
February 18, 2016
In joint work with Emmanuel Kowalski and Philippe Michel, we prove two different estimates on sums of coefficients of modular forms---one related to L-functions and another to the level of distribution. A key step in the argument is a careful analysis of vanishing cycles, a tool originally developed by Lefschetz to study the topology of algebraic varieties. We will explain why this is helpful for these problems.