Journal of the Mathematical Society of Japan

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

Krzysztof KLOSIN

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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 $\phi$.

Article information

J. Math. Soc. Japan, Volume 67, Number 2 (2015), 797-860.

First available in Project Euclid: 21 April 2015

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Primary: 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]
Secondary: 11F55: Other groups and their modular and automorphic forms (several variables) 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F80: Galois representations 11F30: Fourier coefficients of automorphic forms

automorphic forms on unitary groups the Maass lift congruences Bloch–Kato conjecture


KLOSIN, Krzysztof. The Maass space for $U(2,2)$ and the Bloch–Kato conjecture for the symmetric square motive of a modular form. J. Math. Soc. Japan 67 (2015), no. 2, 797--860. doi:10.2969/jmsj/06720797.

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