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

Abstract

Let $K={\bm Q}(i\sqrt{D_K})$ be an imaginary quadratic field of discriminant $-D_K$. We introduce a notion of an adelic Maass space ${\mathcal S}_{k, -k/2}^{\rm M}$ 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 ${\mathcal S}_{k,-k/2}^{\rm M}$ 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 $\ell$ > $k$, we then construct $(\mod \ell)$ congruences between the Maass form corresponding to $\phi$ and Hermitian modular forms orthogonal to ${\mathcal S}_{k,-k/2}^{\rm M}$ whenever ${\rm val}_{\ell}(L^{\rm alg}({\rm Symm}^2 \phi, k))$ > $0$. 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 ${\rm Symm}^2 \rho_{\phi}(k-3)$ and ${\rm Symm}^2 \rho_{\phi}(k)$, where $\rho_{\phi}$ denotes the Galois representation attached to $\phi$.