# Commensurators of abelian subgroups in CAT(0) groups

Tomasz Prytuła

## Institution:

Technical University of Denmark

## Time:

Thursday, February 20, 2020 - 3:00pm

## Location:

RH 306

This talk will introduce work in the area of Geometric Group Theory; no prior background in this area will be assumed. The commensurator of a subgroup H of a group G may be seen as a coarse approximation of the normalizer of H. We consider the situation where H is free abelian and G acts properly on a CAT(0) space, that is, a simply connected space of metric non-positive curvature. The structure of the normalizer of H and its action on the space are well understood in this context. However, the commensurator is more mysterious and it contains subtle information about the action which is not seen by the normalizer. For various classes of CAT(0) spaces we obtain structural results about the commensurator and its relation to the normalizer. In this talk, first I will give background on the commensurator and on CAT(0) spaces and groups, and then I will discuss various geometric tools and constructions used in our approach. This is joint work with Jingyin Huang.

# Waring's Problem via generating functions with nonconstant Fourier coefficients

Garo Sarajian

UCSB

## Time:

Friday, February 28, 2020 - 3:00pm to 4:00pm

## Location:

RH 306

We'll begin by discussing the history of certain problems in Additive Number Theory. Several problems in Additive Number Theory ask how many ways we can represent the elements of a set A as a sum of s elements of the set B, with the two main examples being Waring's Problem and Goldbach's Conjecture. The Hardy-Littlewood Circle Method is the main tool for attacking these problems and often leads to asymptotic formulas for the number of representations. We'll introduce a variant of the Circle Method that simplifies the arguments involved in finding bounds for when the asymptotic formula holds in Waring's Problem.

# $p$-adic estimates for Artin L-functions on curves

## Speaker:

Joe Kramer-Miller

UC Irvine

## Time:

Thursday, February 6, 2020 - 3:00pm to 4:00pm

## Location:

RH 306

Abstract: Let $C$ be a curve over a finite field and let $\rho$ be a nontrivial representation of $\pi_1(C)$. By the Weil conjectures, the Artin $L$-function associated to $\rho$ is a polynomial with algebraic coefficients. Furthermore, the roots of this polynomial are $\ell$-adic units for $\ell \neq p$ and have Archemedian absolute value $\sqrt{q}$. Much less is known about the $p$-adic properties of these roots, except in the case where the image of $\rho$ has order $p$. We prove a lower bound on the $p$-adic Newton polygon of the Artin $L$-function for any representation in terms of local monodromy decompositions. If time permits, we will discuss how this result suggests the existence of a category of wild Hodge modules on Riemann surfaces, whose cohomology is naturally endowed with an irregular Hodge filtration.

# How Quickly Can You Approximate Roots?

Maurice J. Rojas

## Institution:

Texas A&M University

## Time:

Tuesday, January 7, 2020 - 10:00am to 10:50am

## Location:

RH 306

How many digits of an algebraic number A do you need to know before you
are sure you know A? This question dates back to the early 20th century (if
not earlier) and work of Kurt Mahler on the minimal spacing between
complex roots of a degree d univariate polynomial f with integer coefficients of
absolute value at most h: One can bound the minimal spacing explicitly as a function
of d and h. However, the optimality of Mahler's bound for sparse polynomials was open
until recently.

We give a unified family of examples, having just 4 monomial terms, showing
Mahler's bound to be asyptotically optimal over both the p-adic complex numbers,
and the usual complex numbers. However, for polynomials with 3
or fewer terms, we show how to significantly improve Mahler's bound, in both
the p-adic and Archimedean cases. As a consequence, we show how certain
sparse polynomials of degree d can be solved'' in time (log d)^{O(1)} over certain local fields.

This is joint work with Yuyu Zhu.

# Hurwitz tree and equal characteristic deformations of Artin-Schreier covers

Huy Dang

## Institution:

University of Virginia

## Time:

Thursday, January 9, 2020 - 3:00pm to 4:00pm

## Location:

RH 306
An Artin-Schreier curve is a $\mathbb{Z}/p$-branched cover of the projective line over a field of characteristic $p>0$. A unique aspect of positive characteristic is that there exist flat deformations of a wildly ramified cover that change the number of branch points but fix the genus. In this talk, we introduce the notion of Hurwitz tree. It is a combinatorial-differential object that is endowed with essential degeneration data of a deformation. We then show how the existence of a deformation between two covers with different branching data equates to the presence of a Hurwitz tree with behaviors determined by the branching data. One application of this result is to prove that the moduli space of Artin-Schreier covers of fixed genus g is connected when g is sufficiently large. If time permits, we will describe a generalization of the technique for all cyclic covers and the lifting problem.

# Geometry of eigenvarieties for definite unitary groups

Lynnelle Ye

## Institution:

Stanford University

## Time:

Thursday, February 13, 2020 - 3:00pm to 3:50pm

## Location:

RH 306

We will discuss questions about the geometry of Chenevier's eigenvarieties for automorphic forms on definite unitary groups. For example, we will give bounds on the eigenvalues of the $U_p$ Hecke operator that appear in these eigenvarieties. These bounds generalize ones of Liu-Wan-Xiao for rank 2, which they used to prove the Coleman-Mazur-Buzzard-Kilford conjecture in that setting, to all ranks. If time permits, we will discuss possible avenues for recovering additional information not obtainable from these bounds and coming closer to fully generalizing Liu-Wan-Xiao's results.

# Taylor-Wiles-Kisin patching and mod l multiplicities in Shimura curves

Jeff Manning

UCLA

## Time:

Thursday, October 31, 2019 - 3:00pm to 4:00pm

## Location:

RH 440R

In the early 1990s Ribet observed that the classical mod l multiplicity one results for modular curves, which are a consequence of the q-expansion principle, fail to generalize to Shimura curves. Specifically he found examples of Galois representations which occur with multiplicity 2 in the mod l cohomology of a Shimura curve with discriminant pq and level 1.

I will describe a new approach to proving multiplicity statements for Shimura curves, using the Taylor-Wiles-Kisin patching method (which was shown by Diamond to give an alternate proof of multiplicity one in certain cases), as well as specific computations of local Galois deformation rings done by Shotton. This allows us to re-interpret and generalize Ribet's result. I will prove a mod l "multiplicity 2^k" statement in the minimal level case, where k is a number depending only on local Galois theoretic data. This proof also yields additional information the Hecke module structure of the cohomology of a Shimura curve, which among other things has applications to the study of congruence modules.

# Elliptic Curves of Prime Conductor

Aly Deines

## Institution:

Center for Communications Research

## Time:

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

## Location:

RH 440R
The torsion order elliptic curves over $\mathbb{Q}$ with prime conductor have been well studied. In particular, we know that for an elliptic curve $E/\mathbb{Q}$ with conductor $p$ a prime, if $p > 37$, then $E$ has either no torsion, or is a Neumann-Setzer curve and has torsion order 2. In this talk we examine similar behavior for elliptic curves of prime conductor defined over totally real number fields.

# The tame Belyi theorem in positive characteristic

Kiran Kedlaya

UC San Diego

## Time:

Thursday, October 24, 2019 - 3:00pm to 4:00pm

## Location:

RH 440R

If an algebraic curve over a field of characteristic 0 admits a finite
map to the projective line ramified only over three points, then it must
be definable over some number field. This fact has a famous converse due
to Belyi: any curve over a number field admits such a finite map over
its field of definition.

Similarly, if an algebraic curve over a field of characteristic p>0
admits a finite *tamely ramified* map to the projective line ramified
only over three points, then it must be definable over some finite
field. We prove the converse: any curve over a finite field admits such
a finite map over its field of definition.

A construction of Saidi shows that this reduces to the existence of a
single tamely ramified map. This is easy to establish over an infinite
field of odd characteristic, and only slightly harder (using
Poonen-style probabilistic techniques) over a finite field of odd
characteristic. To handle the case of a finite field of characteristic
2, we use a construction of Sugiyama-Yasuda that they used to establish
existence of tame morphisms over an algebraically closed field of
characteristic 2.

Joint work with Daniel Litt (Georgia) and Jakub Witaszek (Michigan).

# Expanding polynomials for sets with additive or multiplicative structure

Cosmin Pohoata

Caltech

## Time:

Thursday, December 5, 2019 - 3:00pm to 4:00pm

## Location:

RH 440R

Abstract: Given an arbitrary set of real numbers A and a two-variate polynomial f with real coefficients, a remarkable theorem of Elekes and R\'onyai from 2000 states that the size |f(A,A)| of the image of f on the cartesian product A x A grows asymptotically faster than |A|, unless f exhibits additive or multiplicative structure. Finding the best quantitative bounds for this intriguing phenomenon (and for variants of it) has generated a lot of interest over the years due to its intimate connection with the sum-product problem from additive combinatorics. In this talk, we will quickly review some of the results in this area, and then discuss some new bounds for |f(A,A)| when the set A has few sums or few products. If time permits, will also discuss some new results over finite fields.