On the curves associated to certain rings of automorphic forms (Q2715664)

Let \(G\) be a reductive algebraic group over \(\mathbb Q\) such that \(G({\mathbb R})\) is compact, and \(U\) an open compact in \(G({\mathbf A}_f)\). The author puts a rational structure on the symmetric space \(X(U) = G({\mathbb Q}) \backslash G({\mathbf A}) / U T({\mathbb R})\), where \(T({\mathbb R})\) is a maximal torus of \(G({\mathbb R})\). For this, he defines the product of two forms of different weights (\(=\) irreducible complex representations), and looks at the ring \(R\) graded by \(n \in {\mathbb Z}\) as follows: fix a highest weight \(\lambda\) representation \(W\), and look only at automorphic forms corresponding to representations with highest weight \(n \lambda\). It turns out that \(R\) is the coordinate ring of a projective embedding of \(X(U)\). Now let \(G(A)=(B \otimes_{\mathbb Q} A)^*/A^*\) for any \({\mathbb Q}\)-algebra \(A\) (where \(B\) is a definite quaternion algebra over \({\mathbb Q}\)), \(\lambda\) correspond to the 3-dimensional adjoint representation of \(G({\mathbb R})\) and \(U = (O \otimes \hat{{\mathbb Z}})^*/\hat{{\mathbb Z}}^*\) for an order \(O\) of \(B\) (first considered by \textit{B. Gross} [Heights and the special values of L-series, CMS Conf. Proc. 7, 115-187 (1987; Zbl 0623.10019)]. Via the Jacquet-Langlands correspondence, one can produce from an automorphic eigenform on \(G\) an automorphic eigenform on \(GL(2)\), to which one can associate a Galois representation (acting on the cohomology of the appropriate moduli space) whose traces of Frobenius are the eigenvalues of the form (by the Eichler-Shimura construction). The starting point of this paper was to find such representations directly in the cohomology of the symmetric space \(X(U)\) without passing through a non-compact group. NEWLINENEWLINENEWLINEHowever, \(X(U)\) is a disjoint union of (orbifold) \({\mathbb P}^1\)'s, so although \(X(U)\) can be given a rational structure, its cohomology is too trivial to carry an interesting action of Galois. The complex points of \(X(U)\) parametrize isomorphism classes of 2-dimensional complex tori \(A={\mathbb C}^2/L\) with a ``proper'' right \(O\)-action, so there is also not enough arithmetic structure in the moduli problem to produce Galois actions. However, such \(A\) are abelian varieties precisely when they have complex multiplication. For such ``algebraically meaningful'' points \(A\), the following analogue of the Eichler-Shimura congruence holds: let \(p\) be a prime where \(O\) is unramified and let \(\phi_p\) denote its Frobenius; let \(\bar{A}\) denote the reduction of \(A\) modulo a fixed prime \(P\) above \(p\); then the action of the Hecke operator on \(A\) has \(p+1\) components, of which \(p\) reduce modulo \(P\) to \(\bar{A}^{\phi^{-1}_p}\), whereas one reduces to \(\bar{A}^{\phi_p}\).