# Reduction of an L-function Modulo an Integer

## Speaker:

Felix Baril Boudreau

## Institution:

University of Western Ontario

## Time:

Thursday, January 13, 2022 - 10:00am to 11:00am

## Location:

https://uci.zoom.us/j/94253918715

Let K be a function field with a constant field of size q. If E is an elliptic curve over K with nonconstant j-invariant then its L-function L(T,E/K) is a polynomia.orgl in 1 + T Z[T]. Inspired by the algorithms of Schoof and Pila for computing zeta functions of curves over finite fields, we consider the problem of computing the reduction of L(T,E/K) modulo an integer without first computing the whole L-function. Doing so for a large enough integer which is coprime with q completely determines L(T,E/K). The existing literature on this problem could be summarized as follows: Under the assumption that the Mordell-Weil group E(K) has a subgroup of order N ≥ 2, with N coprime with q, Chris Hall gave an explicit formula for the reduction L(T,E/K) mod N. We present novel theorems going beyond Hall's. https://arxiv.org/abs/2110.12156

# TBA

Alexander Dunn

Caltech

## Time:

Thursday, February 10, 2022 - 10:00am to 11:00am

Zoom

# TBA

Ben Kane

## Institution:

University of Hong Kong

## Time:

Thursday, February 17, 2022 - 3:00pm to 4:00pm

Zoom

# Unisingular Representations in Arithmetic

John Cullinan

Bard College

## Time:

Thursday, January 27, 2022 - 10:00am to 11:00am

## Location:

Zoom

Let G be a finite group.  A representation V of G is said to be unisingular if det(1-g) = 0 for all g in G.  Unisingular representations arise naturally in arithmetic via point counts on curves over finite fields and l-adic representations on abelian varieties.

In this talk we will survey recent work on properties of elliptic curves and higher dimensional abelian varieties with unisingular l-adic representations with an emphasis on explicit calculation and construction.   Some of this work is joint with John Voight, Jeff Yelton, and Meagan Kenney.

# 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.