Modular Chabauty: Effective S-Integral Point Computation On Curves with Elliptic Fibrations

Abstract

We present a practical, unconditional algorithm for determining the $S$-integral points on any elliptic moduli problem $\mathcal{Y}/\mathbb{Z}[1/S]$ -- that is, on any geometrically connected curve carrying a non-isotrivial elliptic fibration $\mathcal{E} \to \mathcal{Y}$. The associated map $\Phi_M\colon \mathcal{Y} \to \mathcal{M}_{1,1}$ (the modular period map) plays the role ordinarily filled by a $p$-adic period map in Chabauty-type methods. Our Modular Chabauty method studies the image and fibres of $\Phi_M$, and proceeds in two steps: an Effective Shafarevich step, in which we combine the modularity theorem with Cremona's enumeration of elliptic curves by conductor and list all rational elliptic curves with good reduction outside $S$; and a Fibre Computation step, in which we compute the $S$-integral points in the corresponding fibre of $\Phi_M$. A Python/Sage implementation computes $\mathcal{Y}(\mathbb{Z}[1/S])$ for $\mathcal{Y}=\mathbb{P}^1\setminus\{0,1,\infty\}$ and for every modular curve $Y_1(N)$ with $4\le N\le 10$ or $N=12$, for all sets $S$ with $\prod_{p\in S} p^{2}\le 5\cdot 10^{5}$, within $3.5$ seconds on a standard computer.