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Motives over totally real fields and $$p$$-adic $$L$$-functions. (English) Zbl 0808.11034
Summary: Special values of certain $$L$$-functions of the type $$L(M,s)$$ are studied where $$M$$ is a motive over a totally real field $$F$$ with coefficients in another field $$T$$, and $L(M,s)=\prod_{\mathfrak p} L_{\mathfrak p} (M,{\mathcal N}{\mathfrak p}^{-s})$ is an Euler product where $${\mathfrak p}$$ is running through the maximal ideals of the maximal order $${\mathcal O}_ F$$ of $$F$$ and \begin{align*}{ L\sb{\germ p}(M,X)\sp{-1}&=(1-\alpha\sp{(1)} ({\germ p})X)\cdot (1-\alpha\sp{(2)}({\germ p})X)\cdot ... \cdot (1-\alpha (d) ({\germ p})X)\cr &=1+A\sb 1({\germ p})X + ...+ A\sb d({\germ p})X\sp d\cr }\end{align*} being a polynomial with coefficients in $$T$$. Using the Newton and the Hodge polygons of $$M$$ one formulates a conjectural criterion for the existence of a $$p$$-adic analytic continuation of the special values. This conjecture is verified in a number of cases related to Hilbert modular forms.

##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F85 $$p$$-adic theory, local fields 14F20 Étale and other Grothendieck topologies and (co)homologies 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.)
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