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Description

An important tool for bounding the number of rational or torsion points on a curve is to find a function that vanishes at those points and bounding its zeroes. This is essential to Coleman's effective Chabauty and Buium's p-adic differential characters. Buium's work, while appearing quite different, is based on Manin's proof of the function field Mordell conjecture which made use of a Picard-Fuchs differential operator annihilating the periods of an Abelian variety. In this talk, we will discuss a project with Dupuy, Rabinoff, and Zureick-Brown to unify Buium's and Coleman's work in hope of finding more functions vanishing on points of arithmetic interest. This involves constructing differential characters using the p-adic integration theories of Coleman and Colmez, understanding the periods of Abelian varieties, and connections with p-adic Hodge theory.