# Density of rational points on a family of del Pezzo surfaces of degree 1

Rosa Winter

MPI Leipzig

## Time:

Thursday, February 25, 2021 - 10:00am to 11:00am

## Location:

Zoom: https://uci.zoom.us/j/94683355687
Del Pezzo surfaces are surfaces that are classified by their degree $d$, which is an integer between 1 and 9; well-known examples (when $d=3$) are the smooth cubic surfaces in $\mathbb{P}^3$. For del Pezzo surfaces with $d\geq2$ over a field $k$, we know that the set of $k$-rational points is Zariski dense provided that the surface has one $k$-rational point to start with (that lies outside a specific subset of the surface for degree 2). However, for del Pezzo surfaces of degree 1 we do not know if the set of $k$-rational points is Zariski dense in general, even though these surfaces always contain a $k$-rational point. This makes del Pezzo surfaces of degree 1 challenging objects. In this talk I will first explain what del Pezzo surfaces are, and show some of their geometric features, focussing on del Pezzo surfaces of degree 1. I will then talk about a result that is joint work with Julie Desjardins, in which we give necessary and sufficient conditions for the set of $k$-rational points on a specific family of del Pezzo surfaces of degree 1 to be Zariski dense, where $k$ is any field that is finitely generated over $\mathbb{Q}$.

# On the proportion of transverse-free curves

Shamil Asgarli

UBC

## Time:

Thursday, February 18, 2021 - 3:00pm to 4:00pm

## Location:

Zoom: https://uci.zoom.us/j/95368666415

Given a smooth plane curve C defined over an arbitrary field k, we say that C is transverse-free if it has no transverse lines defined over k. If k is an infinite field, then Bertini's theorem guarantees the existence of a transverse line defined over k, and so the transverse-free condition is interesting only in the case when k is a finite field F_q. After fixing a finite field F_q, we can ask the following question: For each degree d, what is the fraction of degree d transverse-free curves among all the degree d curves? In this talk, we will investigate an asymptotic answer to the question as d tends to infinity. This is joint work with Brian Freidin.

# Mod p Galois representations and Abelian varieties

## Speaker:

Shiva Chidambaram

## Institution:

University of Chicago

## Time:

Thursday, January 21, 2021 - 3:00pm to 4:00pm

## Location:

Zoom: https://uci.zoom.us/j/96179378573
The Siegel modular variety $A_2(3)$ which parametrizes abelian surfaces with split level $3$ structure is birational to the Burkhardt quartic threefold. This was shown to be rational over $\mathbb{Q}$ by Bruin and Nasserden. What can we say about its twist $A_2(\rho)$ for a Galois representation \rho valued in $\operatorname{GSp}(4, \mathbb{F}_3)$? While it is not rational in general, it is unirational over $\mathbb{Q}$ by a map of degree at most $6$. In joint work with Frank Calegari and David Roberts, we obtain an explicit description of the universal object over a degree $6$ cover using invariant theoretic ideas. Similar ideas work in other cases, and hence for $(g,p) = (1,2), (1,3), (1,5), (2,2), (2,3)$ and $(3,2)$, any Galois representation $\rho$ valued in $\operatorname{GSp}(2g,\mathbb{F}_p)$ with cyclotomic similitude character arises from the $p$-torsion of a $g$-dimensional abelian variety. When $(g,p)$ is not one of these six tuples, we discuss a local obstruction for representations to arise as torsion.

# The Structure of the Positive Monoid of Integer-Valued Polynomials Evaluated at an Algebraic Number

## Speaker:

Andrei Mandelshtam

UC Irvine

## Time:

Thursday, December 3, 2020 - 3:00pm to 4:00pm

## Location:

Zoom: https://uci.zoom.us/j/95483608618
In the ring $\mathbb{Q}[x]$ of polynomials with coefficients in the rational numbers, it is interesting to consider the subring of all integer-valued polynomials, i.e. polynomial $p(x)$ such that $p(n)$ is an integer for every integer $n$. This ring is known as the most natural and simple example of a non-Noetherian ring. One may wonder whether this is not just the set of all polynomials with integer coefficients. However, e.g. the polynomial $(x^2+x)/2$ is integer-valued. It turns out that this ring consists of exactly the polynomials with integer coefficients in the basis of binomial coefficients $\binom{x}{n}$. Motivated by the characterization of symmetric monoidal functors between Deligne categories, we examine the set $R_{+}(x)$ of polynomials which have nonnegative integer coefficients in this basis. More precisely, we study the set of values of these polynomials at a fixed number $\alpha$. It turns out that this set has a fascinating algebraic structure, explicitly determined by the $p$-adic roots of the minimal polynomial of $\alpha$, which we will fully describe in this talk. This work is joint with Daniil Kalinov, MIT.

# Cokernels of random matrices and distributions of finite abelian p-groups

Nathan Kaplan

UC Irvine

## Time:

Thursday, November 12, 2020 - 3:00pm to 3:50pm

## Location:

Zoom: https://uci.zoom.us/j/95313409309

We will discuss distributions on finite abelian p-groups that arise from taking cokernels of families of random p-adic matrices.  We will explain the motivation for studying these distributions and will highlight several open questions.

# 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.