Local solubility in families of superelliptic curves

Speaker: 

Christopher Keyes

Speaker Link: 

https://www.math.emory.edu/~ckeyes3/

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

Speaker Link: 

https://sieg-baluyot.github.io/

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

Speaker Link: 

https://www.math.ksu.edu/~xiannan/

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

Speaker Link: 

https://personal.math.ubc.ca/~sasgarli/

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

Speaker Link: 

https://public.csusm.edu/ssharif/

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

Speaker Link: 

https://www.hansonsmath.info/

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.

Matrix enumeration over finite fields (Note the special day!)

Speaker: 

Yifeng Huang

Speaker Link: 

https://yifeng-huang-math.github.io/

Institution: 

UBC

Time: 

Tuesday, September 27, 2022 - 3:00pm to 4:00pm

Location: 

RH 306

I will investigate certain matrix enumeration problems over a finite field, guided by the phenomenon that many such problems tend to have a generating function with a nice factorization. I then give a uniform and geometric explanation of the phenomenon that works in many cases, using the statistics of finite-length modules (or coherent sheaves) studied by Cohen and Lenstra. However, my recent work on counting pairs of matrices of the form AB=BA=0 (arXiv: 2110.15566) and AB=uBA for a root of unity u (arXiv: 2110.15570), through purely combinatorial methods, gives examples where the phenomenon still holds true in the absence of the above explanation. Time permitting, I will talk about a partial progress on the system of equations AB=BA, A^2=B^3 in a joint work with Ruofan Jiang. In particular, it verifies a pattern that I previously conjectured in an attempt to explain the phenomenon in the AB=BA=0 case geometrically.

Southern California Number Theory Day

Speaker: 

Aaron Landesman, Michelle Manes, Holly Swisher, Stanley Xiao

Speaker Link: 

https://www.math.uci.edu/~nckaplan/scntd22.html

Institution: 

Harvard University, University of Hawaii, Oregon State University, University of Northern British Columbia

Time: 

Saturday, September 24, 2022 - 9:30am to 5:30pm

Location: 

NS II 1201

Schedule: There will be four one hour invited lectures starting at 10AM and ending around 5:30PM.  A more detailed schedule will be posted soon.

Speakers: Aaron Landesman (Harvard University), Michelle Manes (University of Hawaii), Holly Swisher (Oregon State University), Stanley Xiao (University of Northern British Columbia)

Lightning Talks: We are planning a session where number theory graduate students and postdocs are invited to present their research. These talks will be approximately 5-10 minutes.  If you would like to give a lightning talk, please contact Nathan Kaplan by September 9. Please include your name, affiliation, advisor's name, talk title, and a brief abstract.

Registration: There is no registration fee for the conference, but to help our planning please register.

Location: Natural Sciences II, room 1201 (building 402, located at G6 on this map).

Travel support: Some travel funding is available for participants, with preference given to graduate students and postdocs, especially those giving lightning talks. We also encourage applications from members of under-represented groups. If you would like to apply for funding, please contact Nathan Kaplan with an itemized estimate of expenses, preferably by September 16. Please include your name, affiliation, and advisor's name (if applicable). We strongly encourage carpooling.

Dinner: There will be a conference dinner.  Details TBD.

Degree d points on plane curves

Speaker: 

Lea Beneish

Speaker Link: 

https://sites.google.com/view/lea-beneish/home

Institution: 

UC Berkeley

Time: 

Thursday, October 27, 2022 - 3:00pm to 4:00pm

Location: 

RH 306

Given a plane curve C defined over Q, when the genus of the curve is greater than one, Faltings’ theorem tells us that the set of rational points on the curve is finite. It is then natural to consider higher degree points, that is, points on this curve defined over fields of degree d over Q. We ask for which natural numbers d are there points on the curve in a field of degree d. There is a lot of structure in the set of values d, some of which I will explain in this talk. This talk is based on joint work with Andrew Granville.

A p-adic analogue of an algebraization theorem of Borel

Speaker: 

Abhishek Oswal

Speaker Link: 

https://pma.caltech.edu/people/abhishek-oswal

Institution: 

Caltech

Time: 

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

Location: 

RH 306
Let S be a Shimura variety such that the connected components of the set of complex points $S(\mathbb{C})$ are of the form $D/\Gamma$, where $\Gamma$ is a torsion-free arithmetic group acting on the Hermitian symmetric domain $D$. Borel proved that any holomorphic map from any complex algebraic variety into $S(\mathbb{C})$ is an algebraic map. In this talk I shall describe ongoing joint work with Ananth Shankar and Xinwen Zhu, where we prove a $p$-adic analogue of this result of Borel for compact Shimura varieties of abelian type.

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