Partitions into primes with a Chebotarev condition

Speaker: 

Amita Malik

Institution: 

Max Planck Institute

Time: 

Thursday, May 26, 2022 - 10:00am to 11:00am

Location: 

https://uci.zoom.us/j/95268809663

In this talk, we discuss the asymptotic behavior of the number of integer partitions into primes concerning a Chebotarev condition. In special cases, this reduces to the study of partitions into primes in arithmetic progressions. While the study for ordinary partitions goes back to Hardy and Ramanujan, partitions into primes have been re-visited recently. Our error term is sharp and in the particular case of partitions into prime numbers, we improve on a result of Vaughan. In connection with the monotonicity result of Bateman and Erd\H{o}s, we give an asymptotic formula for the difference of the number of partitions of positive integers which are k-apart.  

Joint distribution of the cokernels of random p-adic matrices

Speaker: 

Jungin Lee

Institution: 

KIAS

Time: 

Thursday, April 14, 2022 - 10:00am to 11:00am

Location: 

Zoom: https://uci.zoom.us/j/91478822337

Let A be a random n by n matrix over Z_p with respect to the Haar measure. Friedman and Washington proved that the distribution of the cokernel of A follows the Cohen-Lenstra distribution. In this talk, we introduce two possible ways to generalize their work. In particular we calculate the joint distribution of the cokernels cok(P_1(A)), ... , cok(P_l(A)) for polynomials P_1(t), ... , P_l(t)∈Z_p[t] under some mild conditions. We also provide a way to understand the linearization of a random matrix model using our result. 

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

Lower Order Term in the Katz-Sarnak Philosophy

Speaker: 

Patrick Meisner

Institution: 

Concordia University

Time: 

Thursday, February 3, 2022 - 10:00am to 11:00am

Location: 

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

Katz and Sarnak predicted that for a nice family of L-functions defined over the ring of polynomials with coefficients in the the field of q elements the Frobenii would become equidistributed in a compact matrix Lie group as q tends to infinity. This talk will discuss the terms which vanish as q tends to infinity for certain statistics of the Frobenii, and shows that for the family of L-functions attached to the r-th power residue symbols, one can describe these lower order terms using random matrix theory.

Bias in cubic Gauss sums: Patterson's conjecture

Speaker: 

Alexander Dunn

Institution: 

Caltech

Time: 

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

Location: 

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

We prove, in this joint work with Maksym Radziwill, a 1978 conjecture of S. Patterson (conditional on the Generalised Riemann hypothesis) concerning the bias of cubic Gauss sums. This explains a well-known numerical bias in the distribution of cubic Gauss sums first observed by Kummer in 1846.

There are two important byproducts of our proof. The first is an explicit level aspect Voronoi summation formula for cubic Gauss sums, extending computations of Patterson and Yoshimoto. Secondly, we show that Heath-Brown's cubic large sieve is sharp under GRH.
This disproves the popular belief that the cubic large sieve can be improved.

An important ingredient in our proof is a dispersion estimate for cubic Gauss sums. It can be interpreted as a cubic large sieve with correction by a non-trivial asymptotic main term. This estimate relies on the Generalised Riemann Hypothesis, and is one of the fundamental reasons why our result is conditional.

Moments of class numbers and distributions of traces of Frobenius in arithmetic progressions

Speaker: 

Ben Kane

Institution: 

University of Hong Kong

Time: 

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

Location: 

Zoom: https://uci.zoom.us/j/94361816690

In this talk, we will discuss the application of modularity and the holomorphic projection operator on a question involving moments for class numbers where the discriminant is restricted to a certain arithmetic progression, with an application to moments for elliptic curves over finite fields where the trace of Frobenius is restricted to the same arithmetic progression.

Unisingular Representations in Arithmetic

Speaker: 

John Cullinan

Institution: 

Bard College

Time: 

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

Location: 

Zoom: https://uci.zoom.us/j/94794065234

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

Speaker: 

Heidi Goodson

Institution: 

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

Speaker: 

Sarah Arpin

Institution: 

University of Colorado

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

Speaker: 

Natalie Evans

Institution: 

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 $h$.

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