p -adic L -functions of Hilbert modular forms

Andrzej Dabrowski

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 4, page 1025-1041
  • ISSN: 0373-0956

Abstract

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We construct p -adic L -functions (in general case unbounded) attached to “motivic" primitive Hilbert cusp forms as a non-archimedean Mellin transform of the corresponding admissible measure. In order to prove the growth conditions of the appropriate complex-valued distributions we represent them as Rankin type representation and use Atkin–Lehner theory and explicit form of Fourier coefficients of Eisenstein series.

How to cite

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Dabrowski, Andrzej. "$p$-adic $L$-functions of Hilbert modular forms." Annales de l'institut Fourier 44.4 (1994): 1025-1041. <http://eudml.org/doc/75088>.

@article{Dabrowski1994,
abstract = {We construct $p$-adic $L$-functions (in general case unbounded) attached to “motivic" primitive Hilbert cusp forms as a non-archimedean Mellin transform of the corresponding admissible measure. In order to prove the growth conditions of the appropriate complex-valued distributions we represent them as Rankin type representation and use Atkin–Lehner theory and explicit form of Fourier coefficients of Eisenstein series.},
author = {Dabrowski, Andrzej},
journal = {Annales de l'institut Fourier},
keywords = {-adic -function; Hilbert cusp form; complex-valued distribution; growth distribution; growth condition; non-archimedean Mellin transform},
language = {eng},
number = {4},
pages = {1025-1041},
publisher = {Association des Annales de l'Institut Fourier},
title = {$p$-adic $L$-functions of Hilbert modular forms},
url = {http://eudml.org/doc/75088},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Dabrowski, Andrzej
TI - $p$-adic $L$-functions of Hilbert modular forms
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 4
SP - 1025
EP - 1041
AB - We construct $p$-adic $L$-functions (in general case unbounded) attached to “motivic" primitive Hilbert cusp forms as a non-archimedean Mellin transform of the corresponding admissible measure. In order to prove the growth conditions of the appropriate complex-valued distributions we represent them as Rankin type representation and use Atkin–Lehner theory and explicit form of Fourier coefficients of Eisenstein series.
LA - eng
KW - -adic -function; Hilbert cusp form; complex-valued distribution; growth distribution; growth condition; non-archimedean Mellin transform
UR - http://eudml.org/doc/75088
ER -

References

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  1. [Dab] A. DABROWSKI, p-adic L-functions of motives and of modular forms I, to be published in Quarterly Journ. Math., Oxford. Zbl0837.11028
  2. [Hi] H. HIDA, On p-adic L-functions of GL(2)×GL(2) over totally real fields, Ann. Inst. Fourier, 40-2 (1991) 311-391. Zbl0739.11019MR93b:11052
  3. [Ka] N.M. KATZ, p-adic L-functions for CM-fields, Invent. Math., 48 (1978) 199-297. Zbl0417.12003MR80h:10039
  4. [Ma] Yu.I. MANIN, Non-Archimedean integration and p-adic L-functions of Jacquet-Langlands, Uspekhi Mat. Nauk, 31 (1976) 5-54 (in Russian). Zbl0336.12007
  5. [Miy] T. MIYAKE, On automorphic forms on GL2 and Hecke operators, Ann. of Math., 94 (1971) 174-189. Zbl0215.37301MR45 #8607
  6. [Pa1] A.A. PANCHISHKIN, Non-Archimedean L-functions associated with Siegel and Hilbert modular forms, Lect. Notes in Math. vol. 1471, Springer-Verlag, 1991. Zbl0732.11026MR93a:11044
  7. [Pa2] A.A. PANCHISHKIN, Motives over totally real fields and p-adic L-functions, Ann. Inst. Fourier, 44-4 (1994). Zbl0808.11034MR96e:11087
  8. [Pa3] A.A. PANCHISHKIN, On non-archimedean Hecke series, In “Algebra” (Ed. by A.I.Kostrikin), Moscow University Press, 1989, 95-141 (in Russian). 
  9. [Pa4] A.A. PANCHISHKIN, p-adic families of motives, Galois representations, and L-functions, preprint MPI, Bonn No.56, 1992. 
  10. [Roh] D.E. ROHRLICH, Nonvanishing of L-functions for GL(2), Inv. Math., 97 (1989) 381-403. Zbl0677.10020MR90g:11062
  11. [Shi] G. SHIMURA, The special values of zeta functions associated with Hilbert modular forms, Duke Math. J., 45 (1978) 637-679. Zbl0394.10015MR80a:10043
  12. [Vi] M.M. VISHIK, Non archimedean measures associated with Dirichlet series, Mat. Sbornik, 99 (1976) 248-260 (in Russian). Zbl0358.14014
  13. [Yo] H. YOSHIDA, On the zeta functions of Shimura varieties and periods of Hilbert modular forms, preprint 1993. 

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