Tohoku Mathematical Journal

On mixed Hodge structures of Shimura varieties attached to inner forms of the symplectic group of degree two

Takayuki Oda and Joachim Schwermer

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We study arithmetic varieties $V$ attached to certain inner forms of $\boldsymbol{Q}$-rank one of the split symplectic $\boldsymbol{Q}$-group of degree two. These naturally arise as unitary groups of a 2-dimensional non-degenerate Hermitian space over an indefinite rational quaternion division algebra. First, we analyze the canonical mixed Hodge structure on the cohomology of these quasi-projective varieties and determine the successive quotients of the corresponding weight filtration. Second, by interpreting the cohomology groups within the framework of the theory of automorphic forms, we determine the internal structure of the cohomology “at infinity” of $V$, that is, the part which is spanned by regular values of suitable Eisenstein series or residues of such. In conclusion, we discuss some relations between the mixed Hodge structure and the so called Eisenstein cohomology. For example, we show that the Eisenstein cohomology in degree two consists of algebraic cycles.

Article information

Tohoku Math. J. (2), Volume 61, Number 1 (2009), 83-113.

First available in Project Euclid: 3 April 2009

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Zentralblatt MATH identifier

Primary: 11F75: Cohomology of arithmetic groups
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35] 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]

Cohomology of arithmetic groups automorphic forms


Oda, Takayuki; Schwermer, Joachim. On mixed Hodge structures of Shimura varieties attached to inner forms of the symplectic group of degree two. Tohoku Math. J. (2) 61 (2009), no. 1, 83--113. doi:10.2748/tmj/1238764548.

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