Function field arithmetic and geometry

Organized by Kevin Chang, Matthew Hase-Liu, and Caleb Ji

This is a learning seminar broadly about problems and techniques using function fields in number theory and algebraic geometry. Examples of such include the Weil conjectures, exponential sums, homological stability, the Cohen-Lenstra heuristics, configuration spaces, the circle method, (more generally) geometric analytic number theory, and hopefully much more!

Schedule

We meet on Mondays from 5:30 to 6:30 in Room 622.

Date Speaker Abstract References
01/29 Kevin Big symplectic or orthogonal monodromy mod l: This talk will be about Hall's 2008 paper of the same name. I'll explain how a simple criterion can be used to prove big mod l monodromy for families of hyperelliptic curves and certain quadratic twist families of elliptic surfaces. I'll also talk about some applications of big mod l monodromy to function field arithmetic (e.g. Cohen-Lenstra heuristics, inverse Galois problem). H06
02/05 Matthew Square-root cancellation for sums of factorization functions over short intervals in function fields: I'll discuss Sawin's paper on estimates for sums of the divisor function (and related arithmetic functions) in short intervals in the function field setting. When the characteristic is large, these estimates approach square-root cancellation and are obtained from bounding the number of points on a certain variety, which are, in turn, controlled by certain cohomology groups. S18
02/12 Caleb The Tate conjecture and BSD for function fields: We give an overview of classical work of Tate and Grothendieck which relate the truth of the Tate conjecture for an elliptic surface over F_q to the finiteness of its Brauer group, BSD for the generic fiber, and the finiteness of the Tate-Shafarevich group of the generic fiber.
02/19 Kevin A degeneration approach to big monodromy: In this talk, I'll explain how big monodromy can be proven using degeneration techniques. I'll cover the examples of families of hyperelliptic and trielliptic curves studied by Achter-Pries.
02/26 Fan Zhou FI-modules (and cohomology of moduli spaces): We roughly follow the famous paper of Church-Ellenberg-Farb on FI-modules and representation stability, with a view towards the Brundan-Stroppel formalism. The speaker’s mathematical abilities permitting, we will provide an application to the cohomology of moduli spaces. CEF12