Organizers
The UCSD number theory group with support from NSF and UCSD.
Invited speakers
Charlotte Chan (University of Michigan), Evan O'Dorney (University of Notre Dame), Congling Qiu (Yale University), Alex Smith (Stanford University)
Location
The conference will be hybrid. The talks will be held in AP&M
Bldg. (Dept. of Math.) Room 6402, on the UCSD campus and/or remotely via Zoom. All talks will be available on Zoom.
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to locate the AP&M Bldg.
(It is the red bldg. on the map.)
For directions and more information, including parking, click here.
Zoom
Registation
If you are planning to come in person, please fill out the registration form by Friday February 4, 2022.
Health and safety
Per campus policy, masks are required indoors for all inperson participants (including speakers), and no food or drink (including water) is permitted in the lecture room. We strongly encourage everyone to be fully vaccinated against COVID19 (including the booster) and to get tested beforehand.
Schedule (all times PST)
9:3010:30am  
Charlotte Chan 
Zoom  
Geometric Lpackets of toral supercuspidal representations


11:00am12:00pm  
Evan O'Dorney 
APM 6402   Reflection theorems for counting quadratic and cubic polynomials


12:002:00pm  
Lunch break



2:003:00pm  
Congling Qiu 
Zoom  
Injectivity of the AbelJacobi map and GrossKudlaSchoen cycles


3:304:30pm  
Alex Smith 
APM 6402  
Simple abelian varieties over finite fields with extreme point counts

Abstracts
Charlotte Chan (University of Michigan)
Geometric Lpackets of toral supercuspidal representations
In 2001, Yu gave an algebraic construction of supercuspidal representations of padic groups. There has since been a lot of progress towards explicitly constructing the local Langlands correspondence for supercuspidal representations: KazhdanVarshavsky and DeBackerReeder (depth zero), Reeder and DeBackerSpice (toral), and Kaletha (regular). In this talk, we present recent and ongoing work investigating a geometric counterpart to this story. This is based on joint work with Masao Oi.
Evan O'Dorney (University of Notre Dame)
Reflection theorems for counting quadratic and cubic polynomials
Scholz's celebrated 1932 reflection principle, relating the 3torsion in the class groups of Q(√D) and Q(√3D), can be viewed as an equality among the numbers of cubic fields of different discriminants. In 1997, Y. Ohno discovered (quite by accident) a beautiful reflection identity relating the number of binary cubic forms, equivalently cubic rings, of discriminants D and 27D, where D is not necessarily squarefree. This was proved in 1998 by Nakagawa, establishing an "extra functional equation" for the Shintani zeta functions counting binary cubic forms. In my talk, I will present a new and more illuminating method for proving identities of this type, based on Poisson summation on adelic cohomology (in the style of Tate's thesis). Also, I will present a corresponding reflection theorem for quadratic polynomials of a quite unexpected shape. The corresponding Shintani zeta function is in two variables, counting by both discriminant and leading coefficient, and finding its analytic properties is a work in progress.

Congling Qiu (Yale University) Injectivity of the AbelJacobi map and GrossKudlaSchoen cycles
On the triple product of a quaternionic Shimura curve over a totally real field, the injectivity of the AbelJacobi map implies an automorphic decomposition of the Chow groups. Then Prasad's theorem on trilinear forms implies the vanishing of the isotypic component of the GrossKudlaSchoen modified diagonal cycle with a certain local root number. We define such a decomposition unconditionally and prove the vanishing. This is a special case of some general results.

Alex Smith (Stanford University)
Simple abelian varieties over finite fields with extreme point counts
Given a compactly supported probability measure on the reals, we will
give a necessary and sufficient condition for there to be a sequence of
totally real algebraic integers whose distribution of conjugates
approaches the measure. We use this result to prove that there are
infinitely many totally positive algebraic integers X satisfying
tr(X)/deg(X) < 1.899; previously, there were only known to be infinitely
many such integers satisfying tr(X)/deg(X) < 2. We also will explain how
our method can be used in the search for simple abelian varieties with
extreme point counts.
