p-adic integrals and linearly dependent points on families of curves I

TL;DR

This paper establishes a linear bound on the distribution of low rank points on families of curves, extending previous exponential bounds and employing advanced methods like Chabauty-Coleman and Ax-Schanuel theorems.

Contribution

It provides a new linear bound for low rank points on families of curves, improving upon existing exponential bounds and applying novel techniques in the field.

Findings

Linear bound for low rank points on families of curves.

Extension of bounds to isotrivial families and solutions to the S-unit equation.

Application of Chabauty-Coleman and Ax-Schanuel methods in this context.

Abstract

We prove that the set of `low rank' points on sufficiently large fibre powers of families of curves are not Zariski dense. The recent work of Dimitrov-Gao-Habegger and K\"uhne (and Yuan) imply the existence of a bound which is exponential in the rank, and the Zilber-Pink conjecture implies a bound which is linear in the rank. Our main result is a (slightly weaker) linear bound for `low ranks'. We also prove analogous results for isotrivial families (with relaxed conditions on the rank) and for solutions to the $S$-unit equation, where the bounds are now sub-exponential in the rank. Our proof involves a notion of the Chabauty-Coleman(-Kim) method in families (or, in some sense, for simply connected varieties). For Zariski non-density, we use the recent work of Bl\`azquez-Sanz, Casale, Freitag and Nagloo on Ax-Schanuel theorems for foliations on principal bundles.