# Sato-Tate Groups in Dimension Greater than 3

Heidi Goodson

Brooklyn College

## Time:

Thursday, January 20, 2022 - 10:00am to 11:00am

## Location:

https://uci.zoom.us/j/92522631731

The focus of this talk is on Sato-Tate groups of abelian varieties -- compact groups predicted to determine the limiting distributions of local zeta functions. In recent years, complete classifications of Sato-Tate groups in dimensions 1, 2, and 3 have been given, but there are obstacles to providing classifications in higher dimensions. In this talk, I will describe my recent work on families of higher dimensional Jacobian varieties. This work is partly joint with Melissa Emory.

# Adding Level Structure to Supersingular Elliptic Curve Isogeny Graphs

Sarah Arpin

## Time:

Thursday, October 14, 2021 - 10:00am to 11:00am

## Location:

Zoom: https://uci.zoom.us/j/91257486031

Supersingular elliptic curves have seen a resurgence in the past decade with new post-quantum cryptographic applications. In this talk, we will discover why and how these curves are used in new cryptographic protocol. Supersingular elliptic curve isogeny graphs can be endowed with additional level structure. We will look at the level structure graphs and the corresponding picture in a quaternion algebra.

# Correlations of almost primes

Natalie Evans

King's College

## Time:

Thursday, October 21, 2021 - 10:00am to 10:50am

## Location:

https://uci.zoom.us/j/95642648816

The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While this conjecture remains wide open, Matom\"{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some recent work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$.

# Sums of two squares are strongly biased towards quadratic residues

Ofir Gorodetsky

## Institution:

Oxford University

## Time:

Thursday, December 2, 2021 - 10:00am to 10:50am

## Location:

https://uci.zoom.us/j/94710132565

Chebyshev famously observed empirically that more often than not, there are more primes of the form 3 mod 4 up to x than primes of the form 1 mod 4. This was confirmed theoretically much later by Rubinstein and Sarnak in a logarithmic density sense. Our understanding of this is conditional on the generalized Riemann Hypothesis as well as Linear Independence of the zeros of L-functions.

We investigate similar questions for sums of two squares in arithmetic progressions. We find a significantly stronger bias than in primes, which happens for almost all integers in a natural density sense. Because the bias is more pronounced, we do not need to assume Linear Independence of zeros, only a Chowla-type Conjecture on non-vanishing of L-functions at 1/2.

We'll aim to be self-contained and define all the notions mentioned above during the talk. We shall review the origin of the bias in the case of primes and the work of Rubinstein and Sarnak. We'll explain the main ideas behind the proof of the bias in the sums-of-squares setting.

# Odd moments in the distribution of primes

Vivian Kuperberg

Stanford

## Time:

Thursday, November 18, 2021 - 3:00pm to 3:50pm

## Location:

https://uci.zoom.us/j/96138712994

In 2004, Montgomery and Soundararajan showed (conditionally) that the distribution of the number of primes in appropriately sized intervals is approximately Gaussian and has a somewhat smaller variance than you might expect from modeling the primes as a purely random sequence. Their work depends on evaluating sums of certain arithmetic constants that generalize the twin prime constant, known as singular series. In particular, these sums exhibit square-root cancellation in each term if they have an even number of terms, but if they have an odd number of terms, there should be slightly more than square-root cancellation. I will discuss sums of singular series with an odd number of terms, including tighter bounds for small cases and the function field analog. I will also explain how this problem is connected to a simple problem about adding fractions.

# Ratios conjecture and multiple Dirichlet series

Martin Cech

## Institution:

Concordia University

## Time:

Thursday, November 4, 2021 - 10:00am to 11:00am

## Location:

https://uci.zoom.us/j/95053211230

Conrey, Farmer and Zirnbauer formulated the ratios conjectures, which give asymptotic formulas for the ratios of products of shifted L-functions from some family. They have many corollaries to other problems in arithmetic statistics, such as the computation of various moments or the distribution of zeros in a family of L-functions.

During the talk, we will show how to use multiple Dirichlet series to prove the conjectures in the family of real Dirichlet L-functions for some range of the shifts. The talk will be accessible even to those with little background in analytic number theory.

# Primes in short intervals - Heuristics and calculations

Allysa Lumley

CRM

## Time:

Thursday, October 28, 2021 - 3:00pm to 3:50pm

## Location:

https://uci.zoom.us/j/99192240652

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length  $y$ around $x$, where $y\ll(\log x)^2$. In particular, we conjecture that the maximum grows surprisingly slowly as $y$ranges from $\log x$ to $(\log x)^2$. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification. This is joint work with Andrew Granville.

# The Paramodular Conjecture for abelian surfaces with rational torsion

Krzysztof Klosin

## Institution:

Queens College, CUNY

## Time:

Thursday, May 13, 2021 - 3:00pm

## Location:

Zoom: https://uci.zoom.us/j/98027654087

# On a universal deformation ring that is a discrete valuation ring

Geoffrey Akers

## Time:

Thursday, May 20, 2021 - 3:00pm

## Location:

Zoom https://uci.zoom.us/j/97940217018

We consider a crystalline universal deformation ring R of an n-dimensional, mod p Galois representation whose semisimplification is the direct sum of two non-isomorphic absolutely irreducible representations. Under some hypotheses, we obtain that R is a discrete valuation ring. The method examines the ideal of reducibility of R, which is used to construct extensions of representations in a Selmer group with specified dimension.  This can be used to deduce modularity of representations.

# Subring growth in integral rings

Sarthak Chimni

## Institution:

University of Illinois, Chicago

## Time:

Thursday, May 27, 2021 - 3:00pm to 4:00pm

## Location:

Zoom: https://uci.zoom.us/j/95840342810

An integral ring R is a ring additively isomorphic to Z^n . The subring zeta function is an important tool in studying subring growth in these rings. One can compute these zeta functions using p-adic integration due to a result of Grunewald, Segal and Smith. I shall talk about computing these zeta functions for Z[t]/(t^n) for small n and describe some results on subring growth and ideal growth for integral rings. This includes joint work with Ramin Takloo-Bighash and Gautam Chinta.