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Zbl 0808.11034
Panchishkin, Alexei A.
Motives over totally real fields and $p$-adic $L$-functions. (English)
[J] Ann. Inst. Fourier 44, No.4, 989-1023 (1994). ISSN 0373-0956; ISSN 1777-5310/e

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 $$L(M,s)=\prod\sb{\germ p} L\sb{\germ p} (M,{\cal N}{\germ p}\sp{-s})$$ is an Euler product where ${\germ p}$ is running through the maximal ideals of the maximal order ${\cal O}\sb F$ of $F$ and $$\eqalign{ L\sb{\germ p}(M,X)\sp{-1}&=(1-\alpha\sp{(1)} ({\germ p})X)\cdot (1-\alpha\sp{(2)}({\germ p})X)\cdot ... \cdot (1-\alpha (d) ({\germ p})X)\cr &=1+A\sb 1({\germ p})X + ...+ A\sb d({\germ p})X\sp d\cr }$$ 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 2000:: *11F67 Special values of automorphic L-series, etc
11F41 Hilbert modular forms and surfaces
11F85 p-adic theory, local fields
14F20 Grothendieck cohomology and topology
11G40 L-functions of varieties over global fields
11S80 Other analytic theory of local fields

Keywords: motives; Newton polygon; Hodge polygon; $p$-adic $L$-function; critical values; periods; Hilbert modular forms; $p$-adic analytic continuation; special values

Cited in: Zbl 0819.11046

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