Restriction of Scalars Chabauty and the $S$-unit equation

Abstract

Given a smooth, proper, geometrically integral curve $X$ of genus $g$ with Jacobian $J$ over a number field $K$, Chabauty's method is a $p$-adic technique to bound $\# X(K)$ when $\mathrm{rank}\ J(K) < g$. We study limitations of a variant called `Restriction of Scalars Chabauty' (RoS Chabauty). RoS Chabauty typically bounds $\# X(K)$ when $\mathrm{rank}\ J(K) \leq [K:\mathbb{Q}] (g - 1)$, but fails in the presence of a subgroup obstruction, a high-rank subgroup scheme of $\mathrm{Res}_{K/\mathbb{Q}} J$ which intersects the image of $\mathrm{Res}_{K/\mathbb{Q}} X$ in higher-than-expected dimension. We define BCP obstructions, which are certain subgroup obstructions arising from the geometry of $X$. BCP obstructions explain all known examples where RoS Chabauty fails to bound $\# X(K)$. We also extend RoS Chabauty to compute $S$-integral points on affine curves. Suppose $K$ does not contain a CM-subfield. We present a $p$-adic algorithm which conjecturally computes solutions to the $S$-unit equation $x+y = 1$ for $x,y \in \mathcal{O}_{K,S}^{\times}$ by using RoS Chabauty to compute $S$-integral points on certain genus $0$ affine curves. As evidence the algorithm succeeds, we prove that all but one of these curves have no subgroup obstructions and that the remaining curve has no BCP obstructions to RoS Chabauty. In contrast, under a generalized Leopoldt conjecture, we prove that analogous methods using classical Chabauty cannot bound solutions to the $S$-unit equation when $[K:\mathbb Q] \geq 3$ and $K$ is not totally real.