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Let K be an imaginary quadratic field of discriminant -D_K 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.

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