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

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.Comment: Comments welcom

Quadratic Chabauty for modular curves:algorithms and examples

Horizon 2020(H2020)101076941Number theory, Algebra and Geometr

Explicit Chabauty--Kim for the Split Cartan Modular Curve of Level 13

We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in an algorithm to compute the rational points on a curve of genus g ≥ 2 over the rationals whose Jacobian has Mordell-Weil rank g and Picard number greater than one, and which satisfies some additional conditions. This algorithm is then applied to the modular curve X_s(13), completing the classification of non-CM elliptic curves over Q with split Cartan level structure due to Bilu–Parent and Bilu–Parent–Rebolledo

Quadratic Chabauty for modular curves: algorithms and examples

We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus g whose Jacobians have Mordell–Weil rank g. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or nontrivial local height contributions away from our working prime. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients X^+_0 (N) of prime level N, the curve X_S4 (13), as well as a few other curves relevant to Mazur’s Program B.https://arxiv.org/abs/2101.01862First author draf