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arXiv · 2206.04304

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

Netan Dogra
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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.

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Netan Dogra. 2022-06-09. p-adic integrals and linearly dependent points on families of curves I. https://arxiv.org/abs/2206.04304

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