Moduli for pairs of elliptic curves with isomorphic \(N\)-torsion (Q5945014)

This paper studies the moduli surfaces \(X_{\simeq}(N)\) for pairs of elliptic curves together with an isomorphism between their \(N\)-torsion groups. These surfaces, which can be seen as degenerate Hilbert modular varieties, are related to classical modular curves. The Weil pairing gives a determinant map from \(X_{\simeq}(N)\) to \((\mathbb{Z}/N\mathbb{Z})^\ast\), whose fibres are the components \(X_{\simeq,\varepsilon}(N)\) of \(X_{\simeq}(N)\). The components have their own modular interpretation. Their structure as complex surfaces have been studied by \textit{C. F. Hermann} [Manuscr. Math. 72, 95-110 (1991; Zbl 0749.14016)] and \textit{E. Kani} and \textit{W. Schanz} [Math. Z. 227, 337-366 (1998; Zbl 0996.14012)]. The paper introduces the spaces of modular forms in these surfaces and the corresponding Hecke algebras, providing multiplicity one theorems for their action on cuspidal subspaces. The geometric complexity of the different components \(X_{\simeq,\varepsilon}(N)\) of \(X_{\simeq}(N)\) is reflected by their geometric genera. The author exhibits the difference in genera of two components as the dimension of the Hecke kernel, a certain subspace of the space of cuspidal forms on \(X_{\simeq,-1}(N)\). This Hecke kernel is then characterized by means of forms with complex multiplication. For prime level \(N=p\), an explicit formula is given for the difference in genera of two components of \(X_{\simeq}(N)\). If \(p\equiv 3 \pmod 4\) an explicit construction is also given of the forms in the Hecke kernel for weight \((2,2)\).