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

2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves II

We give refined methods for proving finiteness of the Chabauty--Coleman--Kim set $X(\mathbb{Q}_2 )_2$, when $X$ is a hyperelliptic curve with a rational Weierstrass point. The main developments are methods for computing Selmer conditions at $2$ and $\infty$ for the mod 2 Bloch--Kato Selmer group associated to the higher Chow group $\mathrm{CH}^2 (\mathrm{Jac}(X),1)$. As a result we show that most genus 2 curves in the LMFDB of Mordell--Weil rank 2 with exactly one rational Weierstrass point satsify $\# X(\mathbb{Q}_2 )_2 <\infty$. We also obtain a field-theoretic description of second descent on the Jacobian of a hyperelliptic curve (under some conditions).Comment: 29 pages. Comments welcome

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