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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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