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The Maass space for \(U(2,2)\) and the Bloch-Kato conjecture for the symmetric square motive of a modular form. (English) Zbl 1396.11071

Summary: Let \(K=\mathbb Q(i\sqrt{D_K})\) be an imaginary quadratic field of discriminant \(-D_K\). We introduce a notion of an adelic Maass space \(\mathcal S_{k, -k/2}^{\mathrm{M}}\) 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 \(\mathcal S_{k,-k/2}^{\mathrm{M}}\) 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 \(\mathcal S_{k,-k/2}^{\mathrm{M}}\) whenever \(\text{val}_{\ell}(L^{\mathrm{alg}}(\mathrm{Symm}^2 \phi, k)) > 0\). 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 \(\mathrm{Symm}^2 \rho_{\phi}(k-3)\) and \(\mathrm{Symm}^2 \rho_{\phi}(k)\), where \(\rho_{\phi}\) denotes the Galois representation attached to \(\phi\).

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms
11F55 Other groups and their modular and automorphic forms (several variables)
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F80 Galois representations
11F30 Fourier coefficients of automorphic forms

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