Southern California Number Theory Day

The UCSD number theory group with support from NSF and UCSD.

Charlotte Chan (University of Michigan), Evan O'Dorney (University of Notre Dame), Congling Qiu (Yale University), Alex Smith (Stanford University)

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.
Click on campus map to locate the AP&M Bldg. (It is the red bldg. on the map.)
For directions and more information, including parking, click here.

We will use the same Zoom coordinates as our Number Theory Seminar. Register via this link to get the Zoom link via email.

If you are planning to come in person, please fill out the registration form by Friday February 4, 2022.

Per campus policy, masks are required indoors for all in-person 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 COVID-19 (including the booster) and to get tested beforehand.

9:30-10:30am		Charlotte Chan
Zoom		Geometric L-packets of toral supercuspidal representations
11:00am-12:00pm		Evan O'Dorney
APM 6402		Reflection theorems for counting quadratic and cubic polynomials
12:00-2:00pm		Lunch break
2:00-3:00pm		Congling Qiu
Zoom		Injectivity of the Abel-Jacobi map and Gross-Kudla-Schoen cycles
3:30-4:30pm		Alex Smith
APM 6402		Simple abelian varieties over finite fields with extreme point counts

• Charlotte Chan (University of Michigan)
  Geometric L-packets of toral supercuspidal representations
  In 2001, Yu gave an algebraic construction of supercuspidal representations of p-adic groups. There has since been a lot of progress towards explicitly constructing the local Langlands correspondence for supercuspidal representations: Kazhdan-Varshavsky and DeBacker-Reeder (depth zero), Reeder and DeBacker-Spice (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.

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• Evan O'Dorney (University of Notre Dame)
  Reflection theorems for counting quadratic and cubic polynomials
  Scholz's celebrated 1932 reflection principle, relating the 3-torsion 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.

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• Congling Qiu (Yale University)
  Injectivity of the Abel-Jacobi map and Gross-Kudla-Schoen cycles
  On the triple product of a quaternionic Shimura curve over a totally real field, the injectivity of the Abel-Jacobi 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 Gross-Kudla-Schoen 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.

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• 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.

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