Gauss composition with level structure, polyharmonic Maass forms, and Hecke series

Speaker: 

Olivia Beckwith

Institution: 

UIUC

Time: 

Thursday, April 8, 2021 - 3:00pm to 4:00pm

Location: 

Zoom: https://uci.zoom.us/j/95973703658

 

The Gauss composition law famously describes the class group of an order in a quadratic number field by an operation on binary quadratic forms up to matrix transformation. Using a stricter notion of equivalence, we describe ray class groups of a quadratic order in terms of quadratic forms.  We explore applications to representing primes by binary quadratic forms, and we describe leading coefficients of Hecke series for real quadratic fields as twisted traces of cycle integrals of polyharmonic Maass forms. This is ongoing joint work with Gene Kopp. 

Local-Global Phenomena for Elliptic Curves

Speaker: 

Jacob Mayle

Institution: 

University of Illinois, Chicago

Time: 

Thursday, March 11, 2021 - 3:00pm to 4:00pm

Location: 

Zoom: https://uci.zoom.us/j/95528784206

 

A local-global principle is a result that allows us to deduce global information about an object from local information. A well-known example is the Hasse-Minkowski theorem, which asserts that a quadratic form represents a number if and only if it does so everywhere locally. In this talk, we'll discuss certain local-global principles in arithmetic geometry, highlighting two that are related to elliptic curves, one for torsion and one for isogenies. In contrast to the Hasse-Minkowski theorem, we'll see that these two results exhibit considerable rigidity in the sense that a failure of either of their corresponding everywhere local conditions must be rather significant.

An isometric invariant of combinatorial type on (F_q^n,x_1^2+...+x_n^2) over F_q

Speaker: 

Semin Yoo

Institution: 

University of Rochester

Time: 

Thursday, January 14, 2021 - 3:00pm to 4:00pm

Location: 

Zoom: https://uci.zoom.us/j/94525934294
In this talk, we introduce a new isometric invariant of combinatorial type on the quadratic space $(\mathbb{F}_{q}^{n},x_{1}^{2}+\cdots+x_{n}^{2})$ over $\mathbb{F}_{q}$. First, we recall some basic facts about quadratic forms. In particular, we will restrict ourselves to the case, where the base field is finite. In order to define this new invariant, we introduce special types of lines, named after line types in Minkowski's geometry. We prove that counting lines of each type is an isometric invariant on the quadratic space $(\mathbb{F}_{q}^{n},x_{1}^{2}+\cdots+x_{n}^{2})$ over $\mathbb{F}_{q}$. In order to prove this theorem, we redrive Minkowski's formula for the size of spheres on finite fields in an elementary way. Only some elementary facts of number theory are required for this talk.

Computing modular forms using supersingular isogeny graphs

Speaker: 

Alex 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

Speaker: 

Fatma Karaoğlu

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

Speaker: 

Rosa 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

Speaker: 

Shamil Asgarli

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

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

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

Speaker: 

Nathan Kaplan

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.

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