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

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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

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Abstract

Let K = Q(i√DK) be an imaginary quadratic field of discriminant −DK. We introduce a notion of an adelic Maass space \mathcal{S}Mk,−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 DK is prime we obtain a Hecke-equivariant descent from \mathcal{S}Mk,−k/2 to the space of elliptic cusp forms Sk−1(DK, χK), where χK is the quadratic character of K. For a given ϕ ∈ Sk−1(DK, χK), 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}Mk,−k/2 whenever valℓ(Lalg(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 Symm2ρϕ(k−3) and Symm2ρϕ(k), where ρϕ denotes the Galois representation attached to ϕ.

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