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On the zeta functions of Shimura varieties and periods of Hilbert modular forms. (English) Zbl 0823.11018
Let $$F$$ be a totally real algebraic number field of degree $$n$$ and let $$B$$ be a quaternion algebra over $$F$$ such that $$B \otimes_ \mathbb{Q} \mathbb{R}\cong M_ 2 (\mathbb{R})^ r \times \mathbb{H}^{n -r}$$ with $$r>0$$, where $$\mathbb{H}$$ is the Hamilton quaternion algebra. Let $$H= \text{Gal} (\overline {\mathbb{Q}}/ F)$$, and let $$\Omega$$ be the subset of $$\text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})/ H$$ consisting of the archimedean places of $$F$$ at which $$B$$ splits. Let $H'= \{g\in \text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})\mid g \Omega= \Omega\},$ and let $$F'$$ be the fixed field of $$H'$$. Let $$G= \text{Res}_{F\wedge \mathbb{Q}} (B')$$, and let $$K$$ be a compact open subgroup of $$G (\mathbb{A}_ f)$$. Then there exists a Shimura variety $$S_ K$$ defined over $$F'$$ whose complex points $$S_ K (\mathbb{C})$$ form an analytic space isomorphic to a space of the form $$G (\mathbb{Q}) \setminus G (\mathbb{A})/ KK_ \infty$$. R. P. Langlands [Can. J. Math. 31, 1121- 1216 (1979; Zbl 0444.14016)] determined the Hasse-Weil zeta function $$Z(s, S_ K/ F')$$ of $$S_ K$$ over $$F'$$ whose main part is of the form $$\prod_ \pi L(s, \pi, r_ 1)^{m(\pi, K)}$$, where $$\pi$$ is an automorphic representation and $$r_ 1$$ is a $$2^ r [F': \mathbb{Q}]$$- dimensional representation of the $$L$$-group $$^ L G= GL (2, \mathbb{C})\rtimes \text{Gal} (\overline {\mathbb{Q}}/ \mathbb{Q})$$ of $$G$$.
In this paper, the author shows that $$L(s, \pi, r_ 1)$$ is equal to $$L(s, \bigotimes_ \Omega \text{ Ind}^{H'}_ H \sigma_ \lambda)$$, where $$\sigma_ \lambda$$ is the $$\lambda$$-adic representation associated to $$\pi$$. He also shows that the transcendental part of the critical values of $$L(s, \pi, r_ 1)$$ can be expressed as a product of a power of $$\pi$$ and the $$Q$$-invariants defined by G. Shimura [Invent. Math. 94, 245-305 (1988; Zbl 0656.10018)] if $$r\geq 2$$.

##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14G35 Modular and Shimura varieties 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
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