Motives over totally real fields and p-adic L-functions
Annales de l'Institut Fourier, Volume 44 (1994) no. 4, pp. 989-1023.

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)=𝔭L𝔭(M,𝒩𝔭-s)

is an Euler product 𝔭 running through maximal ideals of the maximal order 𝒪 F of F and

L𝔭(M,X)-1=(1-α(1)(𝔭)X)·(1-α(2)(𝔭)X)·...·(1-α(d)(𝔭)X)=1+A1(𝔭)X+...+Ad(𝔭)Xd

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.

On étudie des valeurs spéciales des fonctions L de type L(M,s)M est un motif sur un corps totalement réel F à coefficients dans un corps de nombres T, et

L(M,s)=𝔭L𝔭(M,𝒩𝔭-s)

est un produit eulérien étendu sur tous les idéaux maximaux 𝔭 de l’ordre maximal 𝒪 F de F et

L𝔭(M,X)-1=(1-α(1)(𝔭)X)·(1-α(2)(𝔭)X)·...·(1-α(d)(𝔭)X)=1+A1(𝔭)X+...+Ad(𝔭)Xd

est un polynôme à coefficients dans T. À l’aide des polygones de Newton et de Hodge de M on formule des conditions conjecturales de l’existence d’un prolongement p-adique analytique de ces valeurs spéciales. On vérifie cette conjecture dans une série d’exemples liés aux formes modulaires de Hilbert.

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     title = {Motives over totally real fields and $p$-adic $L$-functions},
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Panchishkin, Alexei A. Motives over totally real fields and $p$-adic $L$-functions. Annales de l'Institut Fourier, Volume 44 (1994) no. 4, pp. 989-1023. doi : 10.5802/aif.1424. http://www.numdam.org/articles/10.5802/aif.1424/

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