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

Given an integer $N \geq 3$, 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.