MIT Number Theory Seminar
Tuesday, November 28, 2017 at 4:30pm to 5:30pm
Amnon Besser (Ben-Gurion University/Georgia Tech)
Title: "Vologodsky and Coleman integration on curves with semi-stable reduction"
Abstract: A meromorphic one-form on a curve over a p-adic field with semi-stable reduction has two p-adic integrals associated to it, by Coleman and by Vologodsky. We prove that the Vologodsky integral restricts to Coleman integrals on the rigid subdomains reducing to the components of the smooth part of the special fiber and that on the connecting annuli the differences of these Coleman integrals form a harmonic cochain on the edges of the dual graph of the special fiber. This determines the Vologodsky integral completely. I will explain the behavior of the Vologodsky integral on the connecting annuli which has been observed independently and used, by Stoll, Katz-Rabinoff-Zureick-Brown, in works on global bounds on the number of rational points on curves. I will describe an interesting product on 1-forms used in the proof of the Theorem as well as in work on p-adic height pairings. I will explain the original motivation for this result, which is relevant for the interesting question of generalizing the result to iterated integrals (Joint work with Sarah Zerbes).