Speakers for 2022 (Abstracts below)
Sept 21, 2022: Shichen Tang (SUNY Buffalo)
Oct 19, 2022: Dingxin Zhang (Tsinghua University)
Nov 2, 2022: James Upton (UCSD) + Jeremy Booher (UFL)
Nov 16, 2022: Joe Kramer-Miller (Lehigh)
Nov 30, 2022: Nobuo Tsuzuki
September 21, 2022: Shichen Tang
Title: Arithmetic stability of higher rank Artin--Schreier--Witt towers
Abstract: A conjecture of Daqing Wan states that if a p-adic Lie tower "comes from geometry", then it is expected to have some geometric and arithmetic stability. This talk is related to a special case of this conjecture. Consider a higher rank Artin--Schreier--Witt tower over $P^1$ that is strongly genus stable and ramifies only at one point. We will show that the valuations of reciprocal zeros of zeta functions in this tower are equally distributed on the interval [0,1], and under a slightly stronger assumption, the valuations are unions of several arithmetic progressions. This generalizes results of Kosters--Zhu. Further assuming that the tower is generated from an overconvergent power series, we will also prove a spectral halo property of the corresponding Artin--Schreier--Witt eigenvariety.
October 19, 2022: Dingxin Zhang (Tsinghua University)
Title: Visibility and divisibility of Frobenius eigenvalues
Abstract: I shall explain a comparison theorem between an overconvergent variant of Dwork cohomology and rigid cohomology. It allows us to extract interesting information about Frobenius eigenvalues of compactly supported rigid cohomology via Dwork theory. For instance, combining with the method of Adolphson and Sperber, one could obtain improved lower bounds of q-divisibility of Frobenius eigenvalues of algebraic varieties. This is a joint work with Daqing Wan.
November 2, 2022:
(First speaker) Jeremy Booher (UFL)
Title: Towers of Curves and Motivic Class Groups
Abstract: Let C be a smooth projective curve over a finite field of characteristic p. The p-part of the Jacobian of C is the function field analog of the class group of a number field, but this "motivic class group" includes additional geometric information. For example, it determines the group-scheme structure of the p-torsion of Jac(C), the Ekedahl-Oort type of the curve. This talk will give an introduction to motivic class groups and discuss conjectures with Bryden Cais which propose a new kind of Iwasawa theory for motivic class groups in Z_p towers of curves. The follow-up talk by James Upton will focus on joint work with Joe Kramer-Miller and Bryden Cais which establishes a special case of this conjecture.
(Second speaker) James Upton (UCSD)
Title: Equicharacteristic L-Functions for Z_p-Towers
Abstract: Let k be a finite field of characteristic p. In this talk we discuss a certain "equicharacteristic L-function'' L(T,s) attached to a Z_p-tower of curves over k, which in some sense describes the limiting behavior of the L-functions of finite characters of Z_p. For towers satisfying a certain monodromy condition, our work with Joe Kramer-Miller shows that L(T,s) is an entire function and gives strong estimates for its Newton polygon using the theory of Kosters-Zhu. We then indicate some surprising connections between this L-function and the conjectures discussed in Jeremy's talk. We discuss how these connections allow us to apply Dwork-theoretic techniques to establish special instances of those conjectures.
November 16, 2022:
Joe Kramer-Miller (Lehigh)
Title: The distribution of zeros of the Goss zeta function
Abstract: In the late 70's, David Goss introduced a positive characteristic analogue of the Dedekind zeta function for affine curves. The Goss's zeta function satisfies many remarkable properties, which have analogues (sometimes conjectural) in the number field case. For example, there are class number formulas, transcendence results at `odd' special values, and connections to Bernoulli numbers at `even' special values. One missing piece in this analogy is a suitable Riemann hypothesis. In the early 2000s Goss posited several candidates for a Riemann hypothesis, inspired by calculations of Daqing Wan in the affine line case. Previous evidence for these conjectures was sparse: it was only known for 6 curves in any characteristic besides the affine line. In this talk, I will outline joint work with James Upton where we prove (corrected versions) of Goss's conjectures for ordinary curves. Since ordinaryness is a generic condition, this proves the Riemann hypothesis for `most' curves.
November 30, 2022:
Nobuo Tsuzuki (Tohoku University)
Title: Minimal slope conjecture for overconvergent F-isocrystals
Abstract: K.S.Kedlaya proposed a question called minimal slope conjecture (MSC): If two irreducible overconvergent F-isocrystals on a smooth variety over a perfect field of characteristic p > 0 admit slope filtrations as convergent F-isocrystals and if the first steps of both slope filtrations are isomorphic to each other, then two overconvergent F-isocrystals are isomorphic to each other. In this talk will explain our ideas of proof of MSC in the case of curves over an arbitrary perfect field and in the case of finite base fields. In the case of curves one of the keys is to investigate a relation between generic bounded solutions and the maximal slope quotients of generic Frobenius-differential modules.