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

2022/06/09 by Netan Dogra, Dogra, Netan · 2 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2206.04304

openalex publication_date 2022/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Kühne (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 Blàzquez-Sanz, Casale, Freitag and Nagloo on Ax-Schanuel theorems for foliations on principal bundles.

Cited by

• p-adic approaches to unlikely intersections
• A Buium--Coleman bound for the Mordell--Lang conjecture

Related

• Quadratic Chabauty: p-adic height pairings and integral points on hyperelliptic curves
• Bounds on the number of rational points of curves in families
• Restriction of Scalars Chabauty and the S-unit equation
• Bounds on the Chabauty--Kim Locus of Hyperbolic Curves
• Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-Weil rank