Dwork seminar series

Meeting times for Spring 2023

Wednesday from 5-7pm PST; 8-10pm EST; 5-7pm MST; 8-10am Beijing time

Zoom: https://arizona.zoom.us/j/82753596423

If you wish to be added to the mailing list, please contact C. Douglas Haessig (haessig at math.arizona.edu) or Joe Kramer-Miller (jjk221 at lehigh.edu).

Current list of speakers for Spring 2023

Mar 8, 2023: Richard Crew (UFL)

Mar 22, 2023: Yang Liping (Chengdu Univ. of Tech.) and Zhang Hao (S-T Yau Center of Southeast Uni)

Apr 5, 2023: Kiran Kedlaya (UCSD)

Apr 19, 2023: Chuck Doran (U. of Alberta) and Ursula Whitcher (U. of Mich)

May 3, 2023: John Voight (Dartmouth)

March 8, 2023: Richard Crew (University of Florida)

Title: Duality for p-adic differential equations on curves

Abstract: I will start with a survey of ancient and modern results concerning the behavior of the solutions of a p-adic differential equation on a curve, and the question of the finite dimensionality of the de Rham cohomology of a coherent module with a connection. I will then prove a duality theorem for de Rham cohomology which corresponds to classical Lefschetz duality. This is joint work with Andrea Pulita.

March 22, 2023: Yang Liping (Chengdu Univ. of Tech.) and Zhang Hao (S-T Yau Center of Southeast Uni)

Title: Unit roots of the unit root L-functions

Abstract: As a consequence of Wan's theorem about Dwork's conjecture, the unit root L-function constructing from an algebraic family of L-functions are p-adic meromorphic. A natural question is to ask the roots and poles of the unit root L-functions. In this talk, we will introduce some results about the unit roots, which including the work of Haessig and Sperber, and our recent work about the Kloosterman family.

April 5, 2023: Kiran Kedlaya (UCSD)

Title: Uniformities for F-isocrystals on curves

Slides: https://kskedlaya.org/slides/

Abstract: Let X be a smooth curve over a perfect field of characteristic p. Let E be an overconvergent F-isocrystal on X. We ask, and in some cases answer, a few questions of the following form: what properties of E can be controlled in terms of apparent features of X and E, such as the genus of X, the rank of E, and the wild ramification of E? For example, can one bound the length of the divisor at which the Newton polygon disagrees with the generic Newton polygon?

We also briefly indicate the role played by results of this form in the construction of crystalline companions of l-adic local systems (when the base field is finite).

April 19, 2023: Chuck Doran (U. of Alberta) and Ursula Whitcher (U. of Mich)

Title: Arithmetic implications of mirror symmetry constructions

Abstract: Mirror symmetry predicts surprising geometric correspondences between distinct families of algebraic varieties. We describe methods to extract point-counting information and make arithmetic predictions using classical and modern mirror constructions. We focus on implications of the BHK mirror construction as well as possible generalizations to other forms of mirror symmetry.

May 3, 2023: John Voight (Dartmouth)

Title: Zeta functions of cyclic branched covers from hypergeometric functions, and their degeneration

Abstract: We revisit the arithmetic and geometry of certain one-parameter families of cyclic branched covers over finite fields, inspired by Euler's integral representation of classical hypergeometric functions. Katz showed (and many others have further studied) that factors of their zeta function can be interpreted motivically in terms of finite field hypergeometric sums; and under somewhat strong nondegenerate hypotheses, this method gives a full factorization. We extend this to the general case (allowing arbitrary parameters) by studying degenerations. This is joint work with Tyler Kelly.