Panchishkin: Motives over totally real fields and $p$-adic $L$-functions

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Alexei A. Panchishkin
Motives over totally real fields and $p$-adic $L$-functions
Annales de l'institut Fourier, 44 no. 4 (1994), p. 989-1023, doi: 10.5802/aif.1424
Article PDF | Reviews MR 96e:11087 | Zbl 0808.11034 | 2 citations in Cedram

Résumé - Abstract

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

$$L(M,s)=\prod _{\frak p} L_{\frak p} (M,{\cal N}{\frak p}^{-s})$$

is an Euler product ${\frak p}$ running through maximal ideals of the maximal order ${\cal O}_F$ of $F$ and

$$ L_{\frak p}(M,X)^{-1}& =(1-\alpha ^{(1)} ({\frak p})X)\cdot (1-\alpha ^{(2)}({\frak p})X)\cdot ... \cdot (1-\alpha (d) ({\frak p})X)\\ & =1+A_1({\frak p})X + ...+ A_d({\frak p})X^d$$

being a polynomial with coefficients in $T$. Using the Newton and the Hodge polygons of $M$ one formulate a conjectural criterium 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.

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