Iterated p-adic integration on semistable curves

How do you integrate a 1-form on an algebraic curve over the p-adic numbers? One can integrate locally, but because the topology is totally disconnected, it's not possible to perform analytic continuation. For good reduction curves, this question was answered by Coleman who introduced analytic continuation by Frobenius. For bad reduction curves, there are two notions of integration: a local theory that is easy to compute; and a global single-valued theory that is useful for number theoretic applications. We discuss the relationship between these integration theories, concentrating on the p-adic analogue of Chen's iterated integration which is important for the non-Abelian Chabauty method. We explain how to use combinatorial ideas, informed by tropical geometry and Hodge theory, to compare the two integration theories and outline an explicit approach to computing these integrals. This talk will start from the beginning of the story and requires no background besides some fluency in algebraic geometry and topology. This is joint work with Daniel Litt.

The synchronous discussion for Eric Katz’s talk is taking place not in zoom-chat, but at https://tinyurl.com/2021-05-21-ek (and will be deleted after ~3-7 days).