Local contributions to Quadratic Chabauty functions and derivatives of Vologodsky functions with respect to $log(p)$

Abstract: Quadratic Chabauty is a method for finding rational points on curves using $p$-adic methods. The quadratic Chabauty function is a function on these rational points, usually derived from some $p$-adic height, which is a sum of local terms at finite primes. The main term is the term at $p$ which is a Coleman function, but in order to make the method work one needs to be able to compute the finite list of possible values of the other contributions at primes of bad reduction.

Vologodsky functions are the generalisation of Coleman functions to varieties with bad reduction. In this talk, which is based on ongoing work with Steffen Muller and Padma Srinivasan, I would like to promote the general (and vague) idea that the derivative of a Vologodsky integral with respect to the branch of log parameter $log(p)$ is arithmetically interesting.

As an example I will show how the local contribution above a prime $q$ to a $p$ adic height can be computed by deriving the $q$-adic contribution to a $q$-adic height and use this to obtain a computable formula for this contribution using the work of Katz and Litt. In particular, I will recover a formula of Betts and Dogra for the local contribution to the Quadratic Chabauty function at a prime where the completion is a Mumford curve.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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