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

Alexei A. Panchishkin

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 4, page 989-1023
  • ISSN: 0373-0956

Abstract

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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 ) = ∏ 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 L 𝔭 ( M , X ) - 1 = ( 1 - α ( 1 ) ( 𝔭 ) X ) · ( 1 - α ( 2 ) ( 𝔭 ) X ) · ... · ( 1 - α ( d ) ( 𝔭 ) X ) = 1 + A 1 ( 𝔭 ) X + ... + A d ( 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.

How to cite

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Panchishkin, Alexei A.. "Motives over totally real fields and $p$-adic $L$-functions." Annales de l'institut Fourier 44.4 (1994): 989-1023. <http://eudml.org/doc/75097>.

@article{Panchishkin1994,
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\begin\{\}L(M,s)=\prod \_\{\frak p\} L\_\{\frak p\} (M,\{\cal N\}\{\frak p\}^\{-s\})\end\{\}is an Euler product $\{\frak p\}$ running through maximal ideals of the maximal order $\{\cal O\}_F$ of $F$ and\begin\{align\} L\_\{\frak p\}(M,X)^\{-1\}& =(1-\alpha ^\{(1)\} (\{\frak p\})X)\cdot (1-\alpha ^\{(2)\}(\{\frak p\})X)\cdot \ldots \{\} \cdot (1-\alpha (d) (\{\frak p\})X)\\ & =1+A\_1(\{\frak p\})X + \ldots \{\}+ A\_d(\{\frak p\})X^d\end\{align\}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.},
author = {Panchishkin, Alexei A.},
journal = {Annales de l'institut Fourier},
keywords = {motives; Newton polygon; Hodge polygon; -adic -function; critical values; periods; Hilbert modular forms; -adic analytic continuation; special values},
language = {eng},
number = {4},
pages = {989-1023},
publisher = {Association des Annales de l'Institut Fourier},
title = {Motives over totally real fields and $p$-adic $L$-functions},
url = {http://eudml.org/doc/75097},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Panchishkin, Alexei A.
TI - Motives over totally real fields and $p$-adic $L$-functions
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 4
SP - 989
EP - 1023
AB - 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\begin{}L(M,s)=\prod _{\frak p} L_{\frak p} (M,{\cal N}{\frak p}^{-s})\end{}is an Euler product ${\frak p}$ running through maximal ideals of the maximal order ${\cal O}_F$ of $F$ and\begin{align} L_{\frak p}(M,X)^{-1}& =(1-\alpha ^{(1)} ({\frak p})X)\cdot (1-\alpha ^{(2)}({\frak p})X)\cdot \ldots {} \cdot (1-\alpha (d) ({\frak p})X)\\ & =1+A_1({\frak p})X + \ldots {}+ A_d({\frak p})X^d\end{align}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.
LA - eng
KW - motives; Newton polygon; Hodge polygon; -adic -function; critical values; periods; Hilbert modular forms; -adic analytic continuation; special values
UR - http://eudml.org/doc/75097
ER -

References

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