Weights of Coleman functions and effective Chabauty--Kim

Abstract: The Chabauty--Kim method is a technique for studying the rational points on a curve X using motivic properties of quotients U of the fundamental group of X. For specific quotients U, the method has been made effective in work of Coleman and later by Balakrishnan--Dogra, in the sense that it provides an explicit upper bound on the number of rational points. In this talk, I will discuss a recent project in which I extend these effective results to all quotients U, and give some applications (joint work with David Corwin, in progress) towards uniformity results for higher genus curves. A significant part of the proof, which I will discuss in more detail, lies in defining a notion of "weight" for Coleman analytic functions, and showing, following arguments of Balakrishnan--Dogra, that the number of zeroes of a non-zero Coleman analytic function can be bounded in terms of its weight.

number theory

Audience: researchers in the topic

( slides | video )

Rational Points and Galois Representations

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