Modular Chabauty: Effective S-Integral Point Computation On Curves with Elliptic Fibrations
Sa'ar Zehavi
TL;DR
The paper introduces Modular Chabauty, an unconditional algorithm to compute $S$-integral points on elliptic moduli problems by factorizing the problem through a modular period map $\Phi_{\mathcal{M}}:\mathcal{Y}\to\mathcal{M}_{1,1}$. It splits the computation into an Effective Shafarevich Step, which enumerates all elliptic curves with good reduction outside $S$ via the Modularity Theorem and Cremona’s tables, and a Fibre Computation Step that extracts the $S$-integral fibres for each curve $E$ (i.e., $\Phi_{\mathcal{M}}^{-1}(E)(\mathcal{O}_{K,S})$). The authors present two fibre strategies: a universal geometric approach reducing to solving a univariate polynomial and an $E$-torsion–based moduli-theoretic approach tailored for modular curves like $\mathcal{Y}_1(N)$, with division-polynomial techniques facilitating explicit torsion calculations. Their Python/Sage implementation efficiently handles $\mathcal{Y}=\mathbb{P}^1\setminus\{0,1,\infty\}$ and $\mathcal{Y}_1(N)$ for small $N$ and modest $S$ (under $\prod p^2\lesssim 5\cdot 10^5$), yielding concrete, rapid computations of $S$-integral points and related modular-data. This framework advances the Effective Faltings–Siegel program by delivering a practical, model-agnostic method for a broad class of elliptic moduli problems and providing detailed computational evidence for its efficacy on classical modular curves and the thrice-punctured line.
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 $Φ_M\colon \mathcal{Y} \to \mathcal{M}_{1,1}$ (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 $Φ_M$, 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 $Φ_M$. 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.