Diophantine and tropical geometry, and uniformity of rational points on curves

TL;DR

This paper explores the intersection of combinatorics, tropical geometry, and Diophantine geometry to advance understanding of rational points on curves, emphasizing the role of tropical methods and p-adic integration.

Contribution

It introduces new connections between tropical geometry and Diophantine problems, particularly in understanding uniform bounds for rational points on curves.

Findings

Clarifies how tropical geometry aids in p-adic integration analysis.

Connects combinatorics with Diophantine uniformity conjectures.

Highlights the role of the Chabauty--Coleman method in this context.

Abstract

We describe recent work connecting combinatorics and tropical/non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of Chabauty--Coleman lies at the heart of this connection, and we emphasize the clarification that tropical geometry affords throughout the theory of $p$-adic integration, especially to the comparison of analytic continuations of $p$-adic integrals and to the analysis of zeros of integrals on domains admitting monodromy.