I will present an analog of the Chebotarev Density Theorem which works over local fields. As an application, I will use it to prove a conjecture of Bhargava, Cremona, Fisher, and Gajović. This is joint work with Asvin G and Yifan Wei.

Given a family of abelian covers of P^1 branched at at least four points and a prime p of good reduction, by considering the associated Deligne--Mostow Shimura variety, we obtain lower bounds for the Ekedahl-Oort types, and the Newton polygons, at prime p of the curves in the family. In this paper, we investigate whether such lower bounds are sharp. In particular, we prove sharpeness when the number of branching points is at most five and p sufficiently large. Our result is a generalization under stricter assumptions of Irene Bouw, which proves the analogous statement for the p-rank, and it relies on the notion of Hasse-Witt triple introduced by Ben Moonen.  

Southern California Number Theory Day at UC Irvine

        Saturday, October 7th, 2023

SPEAKERS:

Francesc Castella (UCSB)

Nadia Heninger (UCSD)

Kyle Pratt (BYU)

Carl Wang-Erickson (University of Pittsburgh) 

LOCATION: UC Irvine, Natural Sciences II room 1201

The first lecture will begin at 10:00, and the last will end around 5:15.  There will be a dinner after the lectures, details TBA.  More information is available on the conference web page:

 https://www.math.uci.edu/~nckaplan/scntd23.html

which will be updated when the schedule information and talk titles are available.

LIGHTNING TALKS: We are planning a session with LIGHTNING TALKS 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 15th. Please include your name, affiliation, advisor's name (if you are a graduate student), talk title, and brief abstract.

There are no fees (except for the dinner), but we need to know how many people to plan for, so please register using the link on the conference web page. Please email Nathan Kaplan if you have any questions.

Let F be a finite field with q elements. A (projective) algebraic set over F is the set of common zeros in the projective m-space over F of a bunch of homogeneous polynomials in m+1 variables with coefficients in F. Fix positive integers r, m and d with d < q.  We consider the following question:

What is the maximum number of points in an algebraic set in the projective m-s[space over F given by the vanishing of r linearly independent homogeneous polynomials of degree d with coefficients in F?

The case of a single homogeneous polynomial (or in geometric terms, a projective hypersurface) corresponds to a classical inequality proved by Serre in 1989. For the general case, an elaborate conjecture was made by Tsfasman and Boguslavsky, which was open for almost two decades. Recently significant progress in this direction has been made, and it is shown that while the Tsfasman-Boguslavsky Conjecture is true in certain cases, it can be false in general. Some new conjectures have also been proposed. We will give a motivated outline of these developments. If there is time and interest, we will also explain the close connections of these questions to the problem of counting points of sections of Veronese varieties by linear subvarieties of a fixed dimension, and also to coding theory.

This talk is mainly based on joint works with Mrinmoy Datta and with Peter Beelen and Mrinmoy Datta.