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

We present a practical, unconditional algorithm for determining the $S$-integral points on any elliptic moduli problem $\mathcal{Y}\mathrm{/}\mathbb{Z}[1\mathrm{/}S]$ – that is, on any geometrically connected curve carrying a non-isotrivial elliptic fibration $\mathcal{E}\to \mathcal{Y}$. The associated map ${\mathrm{\Phi }}_M\text{\hspace{0.17em}⁣}:\mathcal{Y}\to {\mathcal{M}}_{1,1}$ (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 ${\mathrm{\Phi }}_M$, 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 ${\mathrm{\Phi }}_M$. A Python/Sage implementation computes $\mathcal{Y}(\mathbb{Z}[1\mathrm{/}S])$ for $\mathcal{Y}={\mathbb{P}}^1\setminus \{0,1,\mathrm{\infty }\}$ 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.