Computing modular forms using supersingular isogeny graphs

Number Theory

Speaker:

Alex Cowan

Speaker Link:

http://people.math.harvard.edu/~cowan/

Institution:

Harvard University

Time:

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

Location:

Zoom: https://uci.zoom.us/j/93364693756

We give an efficient algorithm for computing Fourier expansions of weight 2 cusp forms of prime level. The algorithm is based on Mestre's Methode des Graphes and supersingular isogeny graphs, and was used to greatly expand tables in the LMFDB. We'll also talk briefly about work in progress with Kimball Martin about heuristics for estimating the number of degree 2 cusp forms up to a given level.

Smooth Cubic Surfaces with at Least 9 Lines

Number Theory

Speaker:

Fatma Karaoğlu

Speaker Link:

https://ftmakroglu05.github.io/homepage/

Institution:

Tekirdağ Namık Kemal University

Time:

Thursday, March 4, 2021 - 3:00pm to 4:00pm

Location:

Zoom: https://uci.zoom.us/j/99322295399

A cubic surface is an algebraic variety of degree three in projective three space. We will study cubic surfaces over different fields. We are interested in the number of points and lines on a smooth cubic surface. In this talk, we will focus on smooth cubic surfaces with at least 9 lines. There are three cases with 27, 15 and 9 lines, respectively. We will describe these surfaces in terms of normal forms, each of which involves either 4 or 6 parameters over the given field. Using birational maps, the rational pooints on these normal forms will be described explicitly.

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

Number Theory

Speaker:

Rosa Winter

Speaker Link:

https://personal-homepages.mis.mpg.de/winter/

Institution:

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

Number Theory

Speaker:

Shamil Asgarli

Speaker Link:

https://www.math.ubc.ca/~sasgarli/

Institution:

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

Number Theory

Speaker:

Shiva Chidambaram

Speaker Link:

https://math.uchicago.edu/~shivac/

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

Number Theory

Speaker:

Andrei Mandelshtam

Institution:

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

Number Theory

Speaker:

Nathan Kaplan

Speaker Link:

https://www.math.uci.edu/~nckaplan/

Institution:

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

Number Theory

Speaker:

Tomasz Prytuła

Speaker Link:

https://tomaszprytula.com/

Institution:

Technical University of Denmark

Time:

Thursday, February 20, 2020 - 3:00pm

Host:

Christopher Davis

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

Number Theory

Speaker:

Garo Sarajian

Speaker Link:

https://web.math.ucsb.edu/~gsarajian/

Institution:

UCSB

Time:

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

Host:

Nathan Kaplan

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

Number Theory

Speaker:

Joe Kramer-Miller

Institution:

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.

Number Theory

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