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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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