Southern California Number Theory Day

The UCSD number theory group with support from UCSD.

Kevin Ho (UCSD), Debanjana Kundu (University of Regina, Canada), Jack Sempliner (UCLA), Tamara Veenstra (CCR La Jolla)

The talks will be held in person in AP&M Bldg. (Dept. of Math.) Room 6402.
For directions and more information, click here.

If you are planning to come in person, please fill out the registration form by Wednesday February 18, 2026.

See here for a list of options on campus. Or just follow the locals.

Pizza!
We will have pizza from Regents pizzeria (the good stuff!) and in order to make sure we order enough please fill out the registration form by Wednesday February 18, 2026.

If you are a UCSD affiliate, surely you've had to solve this problem before. This information is mostly for visitors.
You will have to pay either by using the ParkMobile app or pay by credit card (Visa, Mastercard, or American Express) at the nearest pay station. Unless you are a UCSD affiliate you will have to park in a visitor parking spot (marked V) and use the ParkMobile zone 4752. The closest parking structures to AP&M are the following, in order of distance.

• Scholars Parking Structure (Sixth College): V spots on level B1, B2 (map)
• Pangea Parking Structure (Roosevelt College): V spots on level 6, 7 (map)
• Osler Parking Structure (South Parking Structure): V spots on level 3 (map)
• Gilman Parking Structure: V spots on levels 1-3 (map)

• Kevin Ho (UCSD)
  Ogg's conjectures over function fields
  In the early 1970s, Andrew Ogg made several conjectures about the rational torsion points of elliptic curves over ℚ and the Jacobians of modular curves. These conjectures were proved shortly after by Barry Mazur as a consequence of his fundamental study of the arithmetic properties of modular curves and Hecke algebras. In this talk, we review the function field analogues of Ogg's conjectures, their current status, and the methods that have been applied to prove some of these conjectures. The methods are based on the ideas of Mazur and Ogg, but there are interesting differences and technical complications that arise in the function field setting, as well as intriguing possible new directions for generalizations. This is joint work with Cécile Armana and Mihran Papikian.

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• Debanjana Kundu (University of Regina, Canada)
  On the p-ranks of class groups of certain Galois extensions
  Let p be an odd prime, let N be a prime with N ≡ 1 (mod p), and let ζ_p be a primitive p-th root of unity. We study the p-rank of the class group of ℚ (ζ_p, N^1/p) using Galois cohomological methods and obtain an exact formula for the p-rank in terms of the dimensions of certain Selmer groups. Using our formula, we provide a numerical criterion to establish upper and lower bounds for the p-rank, analogous to the numerical criteria provided by F. Calegari-M. Emerton and K. Schaefer-E. Stubley for the p-ranks of the class group of ℚ(N^1/p). In the case p=3, we use Redei matrices to provide a numerical criterion to exactly calculate the 3-rank, and also study the distribution of the 3-ranks as N varies through primes which are 4,7 (mod 9). This is joint work with Ufuoma Asenhesa, Rusiru Gambheera, Enrique Nunez Lon-Wo, and Arshay Sheth.

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• Jack Sempliner (UCLA)
  A geometric Jacquet-Langlands correspondence for Shimura varieties
  Over the past decade there has been a great deal of progress by Helm, Tian-Xiao, and finally Xiao-Zhu on the topic of exceptional Hecke correspondences between Shimura varieties in characteristic p. I will explain a new approach (building on that of Xiao-Zhu) to constructing exotic Hecke correspondences purely in the world of p-adic geometry and the geometry of Igusa stacks, which allows one to relate the cohomology of different Shimura varieties in a great deal of generality and with no restrictions on the level at p. Time permitting, I will discuss how these results, in tandem with some results of Fargues-Scholze, allow one to construct certain cases of a cohomological Jacquet-Langlands correspondence between many Shimura varieties of abelian type. All results are joint with Pol van Hoften.

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• Tamara Veenstra (CCR La Jolla)
  Machine learning L-functions
  We conduct a data-scientific investigation of two types of L-functions, revealing "murmuration" type patterns in both cases. In the first type, rational L-functions, we generalize earlier work using machine learning to study the order of vanishing of L-functions on elliptic curves to this broader family. The second type of L-function we study is that of Maass forms. In this case we study the root number of the L-function. In both cases we find both simple and complex machine learning techniques achieve high accuracy on predicting vanishing order or root number. We also present some interesting results and questions about the dataset revealed by our studies.

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