A $p$-adic approach to rational points on curves

TL;DR

This paper explores a new proof of Mordell's conjecture using $p$-adic Galois representations, offering insights into the finiteness of rational solutions on algebraic curves.

Contribution

It presents a novel proof of Mordell's conjecture based on $p$-adic methods, expanding the toolkit for understanding rational points on curves.

Findings

New proof of Mordell's conjecture using $p$-adic Galois representations

Provides a different perspective from Faltings and Vojta's proofs

Highlights the role of $p$-adic variation in rational point finiteness

Abstract

In 1922, Mordell conjectured the striking statement that for a polynomial equation $f(x,y)=0$, if the topology of the set of complex number solutions is complicated enough, then the set of rational number solutions is finite. This was proved by Faltings in 1983, and again by a different method by Vojta in 1991, but neither proof provided a way to provably find all the rational solutions, so the search for other proofs has continued. Recently, Lawrence and Venkatesh found a third proof, relying on variation in families of $p$-adic Galois representations; this is the subject of the present exposition.