Maass spaces on U(2,2) and the Bloch-Kato conjecture for the symmetric square motive of a modular form

Abstract

Let K be an imaginary quadratic field of discriminant -D_K<0. We introduce a notion of an adelic Maass space S_{k, -k/2}^M for automorphic forms on the quasi-split unitary group U(2,2) associated with K and prove that it is stable under the action of all Hecke operators. When D_K is prime we obtain a Hecke-equivariant descent from S_{k,-k/2}^M to the space of elliptic cusp forms S_{k-1}(D_K, \chi_K), where \chi_K is the quadratic character of K. For a given \phi \in S_{k-1}(D_K, \chi_K), a prime l >k, we then construct (mod l) congruences between the Maass form corresponding to \phi and hermitian modular forms orthogonal to S_{k,-k/2}^M whenever the l-adic valuation of L^{alg}(\Sym^2 \phi, k) is positive. This gives a proof of the holomorphic analogue of the unitary version of Harder's conjecture. Finally, we use these congruences to provide evidence for the Bloch-Kato conjecture for the motives \Sym^2 \rho_{\phi}(k-3) and \Sym^2 \rho_{\phi}(k), where \rho_{\phi} denotes the Galois representation attached to \phi.