A p -adic measure attached to the zeta functions associated with two elliptic modular forms. II

Haruzo Hida

Annales de l'institut Fourier (1988)

  • Volume: 38, Issue: 3, page 1-83
  • ISSN: 0373-0956

Abstract

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Let f = n = 1 a ( n ) q n and g = n = 1 b ( n ) q n be holomorphic common eigenforms of all Hecke operators for the congruence subgroup Γ 0 ( N ) of S L 2 ( Z ) with “Nebentypus” character ψ and ξ and of weight k and , respectively. Define the Rankin product of f and g by 𝒟 N ( s , f , g ) = ( n = 1 ψ ξ ( n ) n k + - 2 s - 2 ) ( n = 1 a ( n ) b ( n ) n - s ) . Supposing f and g to be ordinary at a prime p 5 , we shall construct a p -adically analytic L -function of three variables which interpolate the values 𝒟 N ( + m , f , g ) π + 2 m + 1 < f , f > for integers m with 0 m < k - 1 , by regarding all the ingredients m , f and g as variables. Here f , f is the Petersson self-inner product of f .

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Hida, Haruzo. "A $p$-adic measure attached to the zeta functions associated with two elliptic modular forms. II." Annales de l'institut Fourier 38.3 (1988): 1-83. <http://eudml.org/doc/74808>.

@article{Hida1988,
abstract = {Let $f=\sum ^\{\infty \}_\{n=1\}a(n)q^ n$ and $g=\sum ^\{\infty \}_\{n=1\}b(n)q^ n$ be holomorphic common eigenforms of all Hecke operators for the congruence subgroup $\Gamma _ 0(N)$ of $SL_ 2(\{\bf Z\})$ with “Nebentypus” character $\psi $ and $\xi $ and of weight $k$ and $\ell $, respectively. Define the Rankin product of $f$ and $g$ by\begin\{\} \{\cal D\}\_ N(s,f,g)=(\sum ^\{\infty \}\_\{n=1\}\psi \xi (n)n^\{k+\ell -2s- 2\})(\sum ^\{\infty \}\_\{n\ =1\}a(n)b(n)n^\{-s\}). \end\{\}Supposing $f$ and $g$ to be ordinary at a prime $p\ge 5$, we shall construct a $p$-adically analytic $L$-function of three variables which interpolate the values $\{\{\cal D\}_ N(\ell +m,f,g)\over \pi ^\{\ell +2m+1\}&lt; f,f&gt;\}$ for integers $m$ with $0\le m&lt; k-1,$ by regarding all the ingredients $m$, $f$ and $g$ as variables. Here $\langle f,f\rangle $ is the Petersson self-inner product of $f$.},
author = {Hida, Haruzo},
journal = {Annales de l'institut Fourier},
keywords = {Hecke operators; congruence subgroup; Rankin product; p-adically analytic; p-adic interpolation; special values},
language = {eng},
number = {3},
pages = {1-83},
publisher = {Association des Annales de l'Institut Fourier},
title = {A $p$-adic measure attached to the zeta functions associated with two elliptic modular forms. II},
url = {http://eudml.org/doc/74808},
volume = {38},
year = {1988},
}

TY - JOUR
AU - Hida, Haruzo
TI - A $p$-adic measure attached to the zeta functions associated with two elliptic modular forms. II
JO - Annales de l'institut Fourier
PY - 1988
PB - Association des Annales de l'Institut Fourier
VL - 38
IS - 3
SP - 1
EP - 83
AB - Let $f=\sum ^{\infty }_{n=1}a(n)q^ n$ and $g=\sum ^{\infty }_{n=1}b(n)q^ n$ be holomorphic common eigenforms of all Hecke operators for the congruence subgroup $\Gamma _ 0(N)$ of $SL_ 2({\bf Z})$ with “Nebentypus” character $\psi $ and $\xi $ and of weight $k$ and $\ell $, respectively. Define the Rankin product of $f$ and $g$ by\begin{} {\cal D}_ N(s,f,g)=(\sum ^{\infty }_{n=1}\psi \xi (n)n^{k+\ell -2s- 2})(\sum ^{\infty }_{n\ =1}a(n)b(n)n^{-s}). \end{}Supposing $f$ and $g$ to be ordinary at a prime $p\ge 5$, we shall construct a $p$-adically analytic $L$-function of three variables which interpolate the values ${{\cal D}_ N(\ell +m,f,g)\over \pi ^{\ell +2m+1}&lt; f,f&gt;}$ for integers $m$ with $0\le m&lt; k-1,$ by regarding all the ingredients $m$, $f$ and $g$ as variables. Here $\langle f,f\rangle $ is the Petersson self-inner product of $f$.
LA - eng
KW - Hecke operators; congruence subgroup; Rankin product; p-adically analytic; p-adic interpolation; special values
UR - http://eudml.org/doc/74808
ER -

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