The Maass space for U(2,2) and the Bloch–Kato conjecture for the symmetric square motive of a modular form

Abstract

Let K = Q(i√D_K) be an imaginary quadratic field of discriminant −D_K. We introduce a notion of an adelic Maass space \mathcal{S}^M_k,−k/2 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 \mathcal{S}^M_k,−k/2 to the space of elliptic cusp forms S_k−1(D_K, χ_K), where χ_K is the quadratic character of K. For a given ϕ ∈ S_k−1(D_K, χ_K), a prime ℓ > k, we then construct (mod ℓ) congruences between the Maass form corresponding to ϕ and Hermitian modular forms orthogonal to \mathcal{S}^M_k,−k/2 whenever val_ℓ(L^alg(Symm²ϕ, k)) > 0. 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 Symm²ρ_ϕ(k−3) and Symm²ρ_ϕ(k), where ρ_ϕ denotes the Galois representation attached to ϕ.