An exceptional isomorphism between modular varieties

Abstract

This article aims to give a modular construction of a correspondence between modular varieties whose existence was suspected in [Ge-Ny]. The modular varieties in question are on the one hand a Siegel modular threefold X°, that is a moduli space of 2 dimensional abelian varieties, which is a quotient of the moduli space with level 8 structure and on the other hand the self-product (over the base curve) V o of e → Y 0(8), the universal (smooth) elliptic curve with a cyclic subgroup of order 8. These varieties have smooth compactifications X and V respectively and it is shown in (loc. cit.) that the 2 dimensional Galois representation \( {G_Q} = Gall(\overline Q /Q)\; \to \;Aut({H^3}(X,{Q_t})) \) is a subrepresentation of \( {H^3}(X,{Q_t}) \). This fact can be explained by the theory of lifting of elliptic modular cusp forms to Siegel modular forms, but the Tate conjecture relating algebraic cycles and Galois invariant subspaces in the etale cohomology groups predicts the existence of an algebraic cycle Z on X × V inducing the injection of Galois representations H 3(X) → H 3(V). Using explicit equations, J. Stienstra found a dominant rational map X → V whose graph gives the desired cycle. In this paper we give in fact an isomorphism of stacks (cf. Cor 2.10) which also induces the desired inclusion. It seems however that our construction does not directly work for higher levels, whereas the lifting theory is quite general.