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

top
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 ℓ , respectively. Define the Rankin product of f and g by 𝒟 N ( s , f , g ) = ( ∑ n = 1 ∞ ψ ξ ( n ) n k + ℓ - 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 ( ℓ + m , f , g ) π ℓ + 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 .

How to cite

top

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 -

References

top
  1. [1] N. BOURBAKI, Algèbre, Paris, Hermann, 1970. 
  2. [2] N. BOURBAKI, Algèbre commutative, Paris, Hermann, 1961. 
  3. [3] H. CARAYOL, Représentations cuspidales du groupe linéaire, Ann. Scient. Ec. Norm. Sup., 4e-serie, 17 (1984), 191-225. Zbl0549.22009MR86f:22019
  4. [4] W. CASSELMAN, On some results of Atkin and Lehner, Math. Ann., 201 (1973), 301-314. Zbl0239.10015MR49 #2558
  5. [5] P. DELIGNE, Les constantes des équations fonctionnelles des fonctions L, In "Modular functions of one variables II," Lectures notes in Math., 349 (1973), 501-595. Zbl0271.14011MR50 #2128
  6. [6] P. DELIGNE, Valeurs de fonctions L et périodes d'intégrales, Proc. Symp. Pure Math., 33 (1979), part 2, 313-346. Zbl0449.10022MR81d:12009
  7. [7] S. S. GELBART, Automorphic forms on adele groups, Ann. of Math. Studies No. 83, Princeton, Princeton Univ. Press, 1975. Zbl0329.10018MR52 #280
  8. [8] S. S. GELBART and H. JACQUET, A relation between automorphic representations of GL(2) and GL(3), Ann. Scient. Ec. Norm. Sup., 4e-serie, 11 (1978), 471-542. Zbl0406.10022MR81e:10025
  9. [9] E. HECKE, Theorie der Eisensteinschen Reiben höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik, Abh. Math. Hamb., 5 (1927), 199-224 (Werke No. 24). Zbl53.0345.02JFM53.0345.02
  10. [10] H. HIDA, Congruences of cusp forms and special values of their zeta functions, Inventiones Math., 63 (1981), 225-261. Zbl0459.10018MR82g:10044
  11. [11] H. HIDA, A p-adic measure attached to the zeta functions associated with two elliptic modular forms, I, Inventiones Math., 79 (1985), 159-195. Zbl0573.10020MR86m:11097
  12. [12] H. HIDA, Congruences of cusp forms and Hecke algebras, Séminaire de Théorie des Nombres, Paris, 1983-1984, Progress in Math., 59, 133-146. Zbl0575.10025MR89a:11052
  13. [13] H. HIDA, Iwasawa modules attached to congruences of cusp forms, Ann. Scient. Ec. Norm. Sup., 4e-série, 19 (1986), 231-273. Zbl0607.10022MR88i:11023
  14. [14] H. HIDA, Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms, Inventiones Math., 85 (1986), 545-613. Zbl0612.10021MR87k:11049
  15. [15] H. HIDA, Hecke algebras for GL1 and GL2, Séminaire de Théorie des Nombres, Paris 1984-1985, Progress in Math., 63 (1986), 131-163. Zbl0648.10020MR88i:11078
  16. [16] H. HIDA, Modules of congruence of Hecke algebras and L-functions associated with cusp forms, Amer. J. Math., 110 (1988), 323-382. Zbl0645.10029MR89i:11058
  17. [17] H. JACQUET and R.P. LANGLANDS, Automorphic forms on GL(2), Lecture notes in Math., 114, Berlin-Heidelberg-New York, Springer, 1970. Zbl0236.12010MR53 #5481
  18. [18] H. JACQUET, Automorphic forms on GL(2), II, Lecture notes in Math., 278, Berlin-Heidelberg-New York, Springer, 1972. Zbl0243.12005MR58 #27778
  19. [19] N. M. KATZ, Higher congruences between modular forms, Ann. of Math., 101 (1975), 332-367. Zbl0356.10020MR54 #5120
  20. [20] N. M. KATZ, p-adic interpolation of real analytic Eisenstein series, Ann. of Math., 104 (1976), 459-571. Zbl0354.14007MR58 #22071
  21. [21] S. LANG, Cyclotomic fields, Grad. Texts in Math., 59, Berlin-Hiedelberg-New York, Springer, 1978. Zbl0395.12005MR58 #5578
  22. [22] B. MAZUR, Modular curves and the Eisenstein ideal, Publ. Math. I.H.E.S., 47 (1977), 33-186. Zbl0394.14008MR80c:14015
  23. [23] B. MAZUR and A. WILES, On p-adic analytic families of Galois representations, Compositio Math., 59 (1986), 231-264. Zbl0654.12008MR88e:11048
  24. [24] A. A. PANCISKIN, Le prolongement p-adic analytique des fonctions de L de Rankin, I, C.R. Acad. Sc. Paris, 295 (1982), 51-53, II :idem 227-230. Zbl0501.10028MR83m:10039
  25. [25] B. PERRIN-RIOU, Fonctions L p-adiques associées à une forme modulaire et à un corps quadratique imaginaire, J. London Math. Soc. Zbl0656.10019
  26. [26] J.-P. SERRE, Formes modulaires et fonctions zêta p-adiques, In "Modular functions of one variable III", Lecture notes in Math. 350 (1973), pp. 191-268. Zbl0277.12014MR53 #7949a
  27. [27] G. SHIMURA, Introduction to the arithmetic theory of automorphic functions, Tokyo-Princeton, Iwanami Shoten and Princeton Univ. Press, 1971. Zbl0221.10029
  28. [28] G. SHIMURA, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc., (3) 31 (1975), 79-98. Zbl0311.10029MR52 #3064
  29. [29] G. SHIMURA, On some arithmetic properties of modular forms of one and several variables, Ann. of Math., 102 (1975), 491-515. Zbl0327.10028MR58 #10758
  30. [30] G. SHIMURA, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math., 29 (1976), 783-804. Zbl0348.10015MR55 #7925
  31. [31] G. SHIMURA, On the periods of modular forms, Math. Ann., 229 (1977), 211-221. Zbl0363.10019MR57 #3080
  32. [32] G. SHIMURA, Confluent hypergeometric functions on Tube domains, Math. Ann., 260 (1982), 269-302. Zbl0502.10013MR84f:32040
  33. [33] J. TILOUINE, Un sous-groupe p-divisible de la jacobienne de X1(Npr) comme module sur l'algèbre de Hecke, Bull. Soc. Math. France, 115 (1987), 329-360. Zbl0677.14006MR88m:11043
  34. [33] A. WEIL, Basic number theory, Berlin-Heiderberg-New York, Springer, 1974. Zbl0326.12001MR55 #302

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.