$p$-adic $L$-functions of Hilbert modular forms
Annales de l'Institut Fourier, Volume 44 (1994) no. 4, p. 1025-1041

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.

On construit des $L$-fonctions $p$-adiques (en général non bornées) associées aux formes paraboliques et primitives de Hilbert. Nous écrivons les distributions appropriées aux valeurs complexes comme des représentations intégrales de Rankin et, pour démontrer les conditions de croissance, nous utilisons la théorie d’Aktin–Lehner et la forme explicite des coefficients de Fourier des séries d’Eisenstein.

@article{AIF_1994__44_4_1025_0,
author = {Dabrowski, Andrzej},
title = {$p$-adic $L$-functions of Hilbert modular forms},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {44},
number = {4},
year = {1994},
pages = {1025-1041},
doi = {10.5802/aif.1425},
zbl = {0808.11035},
mrnumber = {96b:11065},
language = {en},
url = {http://www.numdam.org/item/AIF_1994__44_4_1025_0}
}

Dabrowski, Andrzej. $p$-adic $L$-functions of Hilbert modular forms. Annales de l'Institut Fourier, Volume 44 (1994) no. 4, pp. 1025-1041. doi : 10.5802/aif.1425. http://www.numdam.org/item/AIF_1994__44_4_1025_0/

[Dab] A. Dabrowski, p-adic L-functions of motives and of modular forms I, to be published in Quarterly Journ. Math., Oxford. | Zbl 0837.11028

[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. | Numdam | MR 93b:11052 | Zbl 0725.11025 | Zbl 0739.11019

[Ka] N.M. Katz, p-adic L-functions for CM-fields, Invent. Math., 48 (1978) 199-297. | MR 80h:10039 | Zbl 0417.12003

[Ma] Yu.I. Manin, Non-Archimedean integration and p-adic L-functions of Jacquet-Langlands, Uspekhi Mat. Nauk, 31 (1976) 5-54 (in Russian). | Zbl 0348.12016 | Zbl 0336.12007

[Miy] T. Miyake, On automorphic forms on GL2 and Hecke operators, Ann. of Math., 94 (1971) 174-189. | MR 45 #8607 | Zbl 0215.37301

[Pa1] A.A. Panchishkin, Non-Archimedean L-functions associated with Siegel and Hilbert modular forms, Lect. Notes in Math. vol. 1471, Springer-Verlag, 1991. | MR 93a:11044 | Zbl 0732.11026

[Pa2] A.A. Panchishkin, Motives over totally real fields and p-adic L-functions, Ann. Inst. Fourier, 44-4 (1994). | Numdam | MR 96e:11087 | Zbl 0808.11034

[Pa3] A.A. Panchishkin, On non-archimedean Hecke series, In “Algebra” (Ed. by A.I.Kostrikin), Moscow University Press, 1989, 95-141 (in Russian).

[Pa4] A.A. Panchishkin, p-adic families of motives, Galois representations, and L-functions, preprint MPI, Bonn No.56, 1992.

[Roh] D.E. Rohrlich, Nonvanishing of L-functions for GL(2), Inv. Math., 97 (1989) 381-403. | MR 90g:11062 | Zbl 0677.10020

[Shi] G. Shimura, The special values of zeta functions associated with Hilbert modular forms, Duke Math. J., 45 (1978) 637-679. | MR 80a:10043 | Zbl 0394.10015

[Vi] M.M. Vishik, Non archimedean measures associated with Dirichlet series, Mat. Sbornik, 99 (1976) 248-260 (in Russian).

[Yo] H. Yoshida, On the zeta functions of Shimura varieties and periods of Hilbert modular forms, preprint 1993.