Title: On the derivative of a Vologodsky function with respect to log(p)

Abstract: Vologodsky functions are generalizations of Coleman functions to the case of bad reduction. They can be expected to play a similar role in various arithmetic formulas in the bad reduction case that Coleman functions play in the good reduction case. The Vologodsky integration theory depends on the choice of a branch of the p-adic logarithm and it turns out that the derivative of Vologodsky functions with respect to the parameter log(p) contains interesting information.

I will focus on one such case, based on joint work with Padma Srinivasan and Steffen Muller. I will recall (from my talk last year) the theory of p-adic log functions on line bundles, p-adic heights associated to p-adic adelic metrics, and canonical log functions and heights on abelian varieties. Then I will explain in detail the following phenomenon: the derivative with respect to log(q) of a q-adic log function, is a q-adic valuation, which is the q-component for a p-adic height. Thus, the theory of Vologodsky functions gives not just the p-component of the p-adic height, but all components.