# Counting polynomials with a prescribed Galois group

Vlad Matei

Tel-Aviv

## Time:

Thursday, April 7, 2022 - 10:00am to 11:00am

## Location:

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

An old problem, dating back to Van der Waerden, asks about counting irreducible polynomials degree $n$ polynomials with coefficients in the box [-H,H] and prescribed Galois group. Van der Waerden was the first to show that H^n+O(H^{n-\delta}) have Galois group S_n and he conjectured that the error term can be improved to o(H^{n-1}).

Recently, Bhargava almost proved van der Waerden conjecture showing that there are O(H^{n-1+\varepsilon}) non S_n extensions, while Chow and Dietmann showed that there are O(H^{n-1.017}) non S_n, non A_n extensions for n>=3 and n\neq 7,8,10.

In joint work with Lior Bary-Soroker, and Or Ben-Porath we use a result of Hilbert to prove a lower bound for the case of G=A_n, and upper and lower bounds for C_2 wreath S_{n/2} . The proof  for A_n can be viewed, on the geometric side,  as constructing a morphism \varphi from A^{n/2} into the variety z^2=\Delta(f) where each varphi_i is a quadratic form.  For the upper bound for C_2 wreath S_{n/2} we prove a monic version of Widmer's result four counting polynomials with imprimitive Galois group.

# Gaussian distribution of squarefree and B-free numbers in short intervals

## Speaker:

Alexander Mangerel

## Institution:

Durham University

## Time:

Thursday, March 31, 2022 - 10:00am to 11:00am

## Location:

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

(Joint with O. Gorodetsky and B. Rodgers) It is a classical quest in analytic number theory to understand the fine-scale distribution of arithmetic sequences such as the primes. For a given length scale h, the number of elements of a nice'' sequence in a uniformly randomly selected interval $(x,x+h], 1 \leq x \leq X$, might be expected to follow the statistics of a normally distributed random variable (in suitable ranges of $1 \leq h \leq X$).  Following the work of Montgomery and Soundararajan, this is known to be true for the primes, but only if we assume several deep and long-standing conjectures such as the Riemann Hypothesis. In fact, previously such distributional results had not been proven for any (non-trivial) sequence of number-theoretic interest, unconditionally.

As a model for the primes, in this talk I will address such statistical questions for the sequence of squarefree numbers, i.e., numbers not divisible by the square of any prime, among other related sifted'' sequences called B-free numbers. I hope to further motivate and explain our main result that shows, unconditionally, that short interval counts of squarefree numbers do satisfy Gaussian statistics, answering several old questions of R.R. Hall.

# The negative Pell equation and applications

Peter Koymans

## Institution:

Univ. of Michigan

## Time:

Thursday, April 21, 2022 - 10:00am to 11:00am

## Location:

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

In this talk we will study the negative Pell equation, which is the conic $C_D : x^2 - D y^2 = -1$ to be solved in integers $x, y \in \mathbb{Z}$. We shall be concerned with the following question: as we vary over squarefree integers $D$, how often is $C_D$ soluble? Stevenhagen conjectured an asymptotic formula for such $D$. Fouvry and Kluners gave upper and lower bounds of the correct order of magnitude. We will discuss a proof of Stevenhagen's conjecture, and potential applications of the new proof techniques. This is joint work with Carlo Pagano.

# Partitions into primes with a Chebotarev condition

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

Jungin Lee

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

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

Alexander Dunn

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

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

John Cullinan

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.