Moments of the Hurwitz zeta function on the critical line

Speaker: 

Anurag Sahay

Institution: 

University of Rochester

Time: 

Thursday, February 2, 2023 - 10:00am to 11:00am

Location: 

RH 306

The Hurwitz zeta function is a shifted integer analogue of the Riemann zeta function, for shift parameters $0<\alpha\leqslant 1$. We consider the integral moments of the Hurwitz zeta function on the critical line $\Re(s)=\tfrac12$. We will focus on rational shift parameters. In this case, the Hurwitz zeta function decomposes as a linear combination of Dirichlet $L$-functions, which leads us into investigating moments of products of $L$-functions. Using heuristics from random matrix theory, we conjecture an asymptotic of the same form as the moments of the Riemann zeta function. If time permits, we will discuss the case of irrational shift parameters, which will include some joint work with Winston Heap and Trevor Wooley and some ongoing work with Heap.

Counting numerical semigroups by Frobenius number, multiplicity, and depth

Speaker: 

Sean Li

Institution: 

MIT

Time: 

Thursday, January 26, 2023 - 10:00am to 11:00am

Location: 

Zoom: https://uci.zoom.us/j/94729574163

A numerical semigroup Λ is a subset of the nonnegative integers which contains 0, has finite complement, and is closed under addition. We characterize Λ by a number of invariants: the genus g = |N0 \ Λ|, the multiplicity m = min(Λ \ {0}), and the Frobenius number f = max(N0 \ Λ). Recently, Eliahou and Fromentin introduced the notion of depth q = ⌈(f+1)/q⌉. In 1990, Backelin showed that the number of numerical semigroups with Frobenius number f approaches Ci · 2^(f/2) for constants C0 and C1 depending on the parity of f. In this talk, we use Kunz words and graph homomorphisms to generalize Backelin’s result to numerical semigroups of arbitrary Frobenius number, multiplicity, and depth, in particular showing that there are ⌊(q+1)^2/4⌋^(f/(2q-2)+o(f)) semigroups with Frobenius number f and depth q.

Local solubility in families of superelliptic curves

Speaker: 

Christopher Keyes

Institution: 

Emory University

Time: 

Thursday, January 19, 2023 - 3:30pm to 4:30pm

Location: 

Zoom: https://uci.zoom.us/j/95668199292
If we choose at random an integral binary form $f(x, z)$ of fixed degree $d$, what is the probability that the superelliptic curve with equation $C \colon: y^m = f(x, z)$ has a $p$-adic point, or better, points everywhere locally? In joint work with Lea Beneish, we show that the proportion of forms $f(x, z)$ for which $C$ is everywhere locally soluble is positive, given by a product of local densities. By studying these local densities, we produce bounds which are suitable enough to pass to the large $d$ limit. In the specific case of curves of the form $y^3 = f(x, z)$ for a binary form of degree 6, we determine the probability of everywhere local solubility to be 96.94\%, with the exact value given by an explicit infinite product of rational function expressions.

Twisted $2k$th moments of primitive Dirichlet $L$-functions: beyond the diagonal

Speaker: 

Siegfred Baluyot

Institution: 

AIM

Time: 

Thursday, December 1, 2022 - 3:00pm to 4:00pm

Location: 

RH 306

In this joint work with Caroline Turnage-Butterbaugh, we study the family of Dirichlet $L$-functions of all even primitive characters of conductor at most $Q$, where $Q$ is a parameter tending to infinity. We approximate the twisted $2k$th moment of this family using Dirichlet polynomials of length between $Q$ and $Q^2$. Assuming the Generalized Lindelof Hypothesis, we prove an asymptotic formula for these approximations. Our result agrees with the prediction of Conrey, Farmer, Keating, Rubinstein, and Snaith, and provides the first rigorous evidence beyond the diagonal terms for their conjectured asymptotic formula for the general $2k$th moment of this family. The main device we use in our proof is the asymptotic large sieve developed by Conrey, Iwaniec, and Soundararajan. 

Quadratic twists of modular L-functions

Speaker: 

Xiannan Li

Institution: 

Kansas State University

Time: 

Thursday, January 12, 2023 - 10:00am to 11:00am

Location: 

https://uci.zoom.us/j/95304920920

The behavior of quadratic twists of modular L-functions is at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves.  Here we describe a proof of an asymptotic for the second moment of this family of L-functions, previously available conditionally on the Generalized Riemann Hypothesis by the work of Soundararajan and Young.  Our proof depends on deriving an optimal large sieve type bound.

Blocking sets arising from plane curves over finite fields

Speaker: 

Shamil Asgarli

Institution: 

Santa Clara University

Time: 

Thursday, October 6, 2022 - 3:00pm to 4:00pm

Location: 

RH 306

Let F_q be a finite field, and consider the set P^2(F_q) of all F_q-points in the projective plane. A subset B of P^2(F_q) is called a blocking set if B meets every line defined over F_q. Given an algebraic plane curve C in P^2, when does the set of F_q-rational points on C form a blocking set? We will see that curves of low degree do not give rise to blocking sets. As an example of this principle, we will show that cubic plane curves defined over F_q do not give rise to blocking sets whenever q is at least 5. On the other hand, we will describe explicit constructions of smooth plane curves (of large degree) that do give rise to blocking sets. Finding blocking curves of optimal degree over a given finite field remains open. This is joint work with Dragos Ghioca and Chi Hoi Yip.

Quantum money from quaternion algebras

Speaker: 

Shahed Sharif

Institution: 

Cal State University, San Marcos

Time: 

Thursday, October 13, 2022 - 3:00pm to 4:00pm

Location: 

RH 306

Public key quantum money is a replacement for paper money which has cryptographic guarantees against counterfeiting. We propose a new idea for public key quantum money. In the abstract sense, our bills are encoded as a joint eigenstate of a fixed system of commuting unitary operators. We show that the proposal is secure against black box attacks. In order to instantiate this protocol, one needs to find a cryptographically complicated system of computable, commuting, unitary operators. To fill this need, we propose using Brandt operators acting on the Brandt modules associated to certain quaternion algebras. This is joint work with Daniel Kane and Alice Silverberg.

Radical Geometric Monogenicity

Speaker: 

Hanson Smith

Institution: 

Cal State University, San Marcos

Time: 

Thursday, November 3, 2022 - 3:00pm to 4:00pm

Location: 

RH 306
A number field is $\textit{monogenic}$ (over $\mathbb{Q}$) if the ring of integers admits a power integral basis, i.e., a $\mathbb{Z}$-basis of the form $\{1, \alpha, \alpha^2,\dots, \alpha^{n-1}\}$. In this case we call $\alpha$ a $\textit{monogenerator}$. The first portion of the talk will be spent revisiting some classical examples of monogenicity and non-monogenicity. We will touch on some recent work on radical extensions while paying particular attention to obstructions to monogenicity and relations to other arithmetic questions. The latter part of the talk will be devoted to recent work ([1] and [2]) constructing a general moduli space of monogenerators. Specifically, given an extension of algebras $B/A$, we construct a $\textit{scheme}$ $\mathcal{M}_{B/A}$ parameterizing the possible choices of a monogenerator for $B$ over $A$. $$ \ $$ $$\textbf{References} $$ [1] Arpin, S., Bozlee, S., Herr, L., and Smith, H. (2021). The Scheme of Monogenic Generators I: Representability. arXiv: https://arxiv.org/abs/2108.07185. (Accepted to Research in Number Theory.) $$ \ $$ [2] Arpin, S., Bozlee, S., Herr, L., and Smith, H. (2022). The Scheme of Monogenic Generators II: Local Monogenicity and Twists. arXiv: https://arxiv.org/abs/2205.04620.

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