# The Weyl law for algebraic tori

Ian Petrow

ETH Zurich

## Time:

Wednesday, May 1, 2019 - 3:00pm to 4:00pm

## Location:

RH 440R

A basic but difficult question in the analytic theory of automorphic forms is: given a reductive group G and a representation r of its L-group, how many automorphic representations of bounded analytic conductor are there? In this talk I will present an answer to this question in the case that G is a torus over a number field.

# Factorization of Hasse-Weil zeta functions of Dwork surfaces

Lian Duan

## Institution:

Univ. of Mass, Amherst

## Time:

Thursday, April 11, 2019 - 3:00pm to 4:00pm

## Location:

RH 306

As a classical example of K3 surfaces, the Dwork surface family is of interest in algebraic geometry and number theory. A lot of work has been done to understand the Hasse-Weil zeta functions of these surfaces. Recent works show that people can totally determine the algebraic part of the zeta function for a general Dwork surface. In this talk, we discuss how to use geometric method to find the explicit factorization of the algebraic part.

# Iwasawa theory for function fields

Bryden Cais

## Institution:

University of Arizona

## Time:

Thursday, June 6, 2019 - 3:00pm to 3:50pm

## Location:

RH 306

Let X_n be a Z_p-tower of smooth projective curves over a
perfect field $k$ of characteristic p that totally ramifies over a finite,
nonempty set of points of X_0 and is unramified elsewhere. In analogy
with the case of number fields, Mazur and Wiles studied the growth of
the p-parts of the class groups $Jac(X_n)[p^infty](\overline{k})$ as n-varies, and
proved that these naturally fit together to yield a module that is
finite and free over the Iwasawa algebra. We introduce a novel
perspective by proposing to study growth of the full p-divisible group
$G_n:=Jac(X_n)[p^infty]$, which may be thought of as the p-primary part of
the *motivic class group* $Jac(X_n)$. One has a canonical decomposition
$G_n = G_n^{et} \times G_n^{m} \times G_n^{ll}$ of $G$ into its etale, multiplicative, and
local-local components, as well as an equality $G_n(\overline{k}) = G_n^{et}(\overline{k})$.

Thus, the work of Mazur and Wiles captures the etale part of G_n, so
also (since Jacobians are principally polarized) the multiplicative
part: both of these p-divisible subgroups satisfy the expected
structural and control theorems in the limit. In contrast, the
local-local components G_n^{ll} are far more mysterious (they can not be
captured by $\overline{k}$-points), and indeed the tower they form has no analogue
in the number field setting. This talk will survey this circle of ideas,
and will present new results and conjectures on the behavior of the
local-local part of the tower G_n.

# ABC Triples in Families

Edray Goins

Pomona College

## Time:

Thursday, April 18, 2019 - 3:00pm to 4:00pm

## Location:

RH 306

Given three positive, relative prime integers A, B, and C such that the first two sum to the third i.e. A + B = C, it is rare to have the product of the primes p dividing them to be smaller than each of the three.  In 1985, David Masser and Joseph Osterlé made this precise by defining a "quality" q(P) for such a triple of integers P = (A,B,C); their celebrated "ABC Conjecture" asserts that it is rare for this quality q(P) to be greater than 1 -- even through there are infinitely many examples where this happens.  In 1987, Gerhard Frey offered an approach to understanding this conjecture by introducing elliptic curves.  In this talk, we introduce families of triples so that the Frey curve has nontrivial torsion subgroup, and explain how certain triples with large quality appear in these families.  We also discuss how these families contain infinitely many examples where the quality q(P) is greater than 1.  This joint work with Alex Barrios.

# Graphs of Hecke operators and Hall algebras

## Speaker:

Roberto Alvarenga

## Institution:

USP-Brazil, visiting UCI

## Time:

Thursday, February 14, 2019 - 3:00pm to 4:00pm

## Location:

RH 306

Abstract: Motivated by questions of Zagier, Lorscheid develops in his Ph. D thesis the theory of graphs of Hecke operators. These graphs encode the action of Hecke operators on automorphic forms over a global function field. In this talk, I intend to present this theory and connect it with the Hall algebras theory. This connection allow us to describe these graphs for an elliptic curve.

# Rank 2 local systems and abelian varieties

## Speaker:

Raju Krishnamoorthy

## Institution:

University of Georgia

## Time:

Wednesday, February 6, 2019 - 3:00pm

## Location:

RH 306

Let X/k be a smooth variety over a finite field. Motivated by work of Corlette-Simpson over the complex numbers, we formulate a conjecture that certain rank 2 local systems on X come from families of abelian varieties. After an introduction to l/p-adic companions, we explain how the existence of a complete set of p-adic companions can be used to approach the conjecture. We also prove Lefschetz theorems for families of abelian varieties over F_q, analogous to work of Simpson over C. This is joint work with A. Pál.

# Higher Eisenstein Congruences

Catherine Hsu

## Institution:

University of Bristol

## Time:

Thursday, May 2, 2019 - 3:00pm

## Location:

RH 306

In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal. In this talk, we re-examine Eisenstein congruences, incorporating a notion of "depth of congruence," in order to understand the local structure of Eisenstein ideals associated to weight 2 cusp forms of squarefree level.

# Multiplicative functions over F_q[X] and the Erdos Discrepancy problem.

Oleksiy Klurman

KTH

## Time:

Thursday, February 28, 2019 - 3:00pm to 4:00pm

We discuss analog of several classical results about mean values of multiplicative functions over F_q[X] explaining some features that are not present in the number field setting. In the first part of the talk, which is based on the joint work with C. Pohoata and K. Soundararajan, we describe spectrum of multiplicative functions over F_q[x]. In the second part of the talk (based on the joint work with A. Mangerel and J. Teravainen), we will focus on the "corrected" function field analog of the Erdos Discrepancy Problem.

# Low degree points on curves

Isabel Vogt

MIT

## Time:

Wednesday, February 27, 2019 - 3:00pm to 4:00pm

## Location:

RH 306

In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite.  By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4.  We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith.

# Random matrices over finite fields follow the Cohen-Lenstra distribution

GilYoung Cheong

## Institution:

University of Michigan

## Time:

Thursday, March 7, 2019 - 3:00pm to 4:00pm

## Location:

RH 306

Given a random n x n matrix over the finite field of p elements for a fixed prime p, we will compute the probability that its Jordan normal form contains a specified 0-Jordan block. As n goes to infinity, we will see that our answer converges to the Cohen-Lenstra distribution, which is conjectured to compute the probability that the class group of a random imaginary quadratic field has a specified p-part (when p is odd). We will see why this happens by making connections between our statistics of random matrices and a heuristic distribution of finite abelian groups given by Cohen and Lenstra.

Much of the talk is from a joint work with Yifeng Huang and Zhan Jiang. We will not assume any background from the audience beyond basic graduate algebra classes.