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

Panchishkin, Alexei A.

doi:10.5802/aif.1424

Motives over totally real fields and

p

-adic

L

-functions

Panchishkin, Alexei A.

Annales de l'Institut Fourier, Volume 44 (1994) no. 4, pp. 989-1023.

Abstract
Résumé

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_{𝔭} L_{𝔭} (M, 𝒩 𝔭^{- s})

is an Euler product $𝔭$ running through maximal ideals of the maximal order $𝒪_{F}$ of $F$ and

\begin{matrix} L_{𝔭} (M, X)^{- 1} & = (1 - α^{(1)} (𝔭) X) \cdot (1 - α^{(2)} (𝔭) X) \cdot ... \cdot (1 - α (d) (𝔭) X) \\ = 1 + A_{1} (𝔭) X + ... + A_{d} (𝔭) X^{d} \end{matrix}

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 $L (M, s)$ où $M$ est un motif sur un corps totalement réel $F$ à coefficients dans un corps de nombres $T$ , et

L (M, s) = \prod_{𝔭} L_{𝔭} (M, 𝒩 𝔭^{- s})

est un produit eulérien étendu sur tous les idéaux maximaux $𝔭$ de l’ordre maximal $𝒪_{F}$ de $F$ et

\begin{matrix} L_{𝔭} (M, X)^{- 1} & = (1 - α^{(1)} (𝔭) X) \cdot (1 - α^{(2)} (𝔭) X) \cdot ... \cdot (1 - α (d) (𝔭) X) \\ = 1 + A_{1} (𝔭) X + ... + A_{d} (𝔭) X^{d} \end{matrix}

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.

MR Zbl | 2 citations in Numdam

DOI: 10.5802/aif.1424

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     author = {Panchishkin, Alexei A.},
     title = {Motives over totally real fields and $p$-adic $L$-functions},
     journal = {Annales de l'Institut Fourier},
     pages = {989--1023},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {44},
     number = {4},
     year = {1994},
     doi = {10.5802/aif.1424},
     mrnumber = {96e:11087},
     zbl = {0808.11034},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1424/}
}

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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://archive.numdam.org/articles/10.5802/aif.1424/

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