Compactified moduli spaces and Hecke correspondences for elliptic curves with a prescribed $N$-torsion scheme

Abstract

Given an integer $N\ge 3$, we prove that for any ring $R$ and any finite locally free commutative $R$-group scheme $G$ whose geometric fibres are isomorphic to the $N$-torsion subscheme of some elliptic curve $E$, there is a smooth affine curve $Y_G(N)$ parametrizing elliptic curves over $R$-schemes whose $N$-torsion subscheme is isomorphic to $G$. We also describe compactifications $X_G(N)$ of these curves when $R$ is a regular excellent Noetherian ring in which $N$ is invertible, as well as construct the Hecke correspondences they are endowed with. As an application, we show that the equations for $X_G(N)$ found over base fields for $N=7,8,9,11,13$ (by Halberstadt--Kraus, Poonen--Schaefer--Stoll, Chen and Fisher) are in fact valid over regular excellent Noetherian bases that are $\mathbb{Q}$-algebras. Finally, we describe in detail the equivalence of this construction with the point of view of Galois twists that these authors use.

Second version (63 p.), the text was largely rewritten. Comments welcome! arXiv admin note: substantial text overlap with arXiv:2501.01315 [author's note: the article is indeed largely based on a chapter of my thesis.]