Compactified moduli spaces and Hecke correspondences for elliptic curves with a prescribed $N$-torsion scheme
Elie Studnia
TL;DR
The paper develops a comprehensive, base-change–friendly framework for moduli of elliptic curves with prescribed $N$-torsion schemes, introducing $Y_G(N)$ and its compactification $X_G(N)$ and equipping them with degeneracy maps and Hecke correspondences. It proves representability and smoothness properties for these moduli spaces over regular/excellent Noetherian bases, and shows that equations for $X_E^{\alpha}(N)$ known over fields extend to these bases, including primes dividing $N$, by a Galois-twists viewpoint. A determinant pairing to the bilinear alternating Pol$(G)$ (Weil pairing) is constructed, enabling a polarized-étale theory and Hecke actions on Jacobians via $T_q$. The results unify the moduli-theoretic, base-change, and degeneracy-operator viewpoints, providing a robust toolkit for studying rational points, reduction behavior, and Galois twists of elliptic curves with prescribed level-$N$ torsion data.
Abstract
Given an integer $N \geq 3$, we prove that for any ring $R$ and any finite locally free $R$-group scheme $G$ which is fppf-locally (over $R$) isomorphic the $N$-torsion subscheme of some elliptic curve $E/R$, 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.