p-adic interpolation of convolutions of Hilbert modular forms
Annales de l'Institut Fourier, Tome 47 (1997) no. 2, pp. 365-428.

Dans cet article nous construisons des mesures p-adiques reliées aux valeurs des convolutions des formes modulaires de Hilbert de poids entier et demi-entier aux points critiques négatifs à condition que le corps de nombre totalement réel F ait un nombre de classes h F =1. Le résultat est parallèle à celui de Panchishkin [Lecture Notes in Math., 1471, Springer Verlag, 1991], qui a traité le cas où les deux formes modulaires ont un poids entier. Pour pouvoir définir les mesures, il nous faut d’abord introduire un opérateur twist et une involution j sur l’espace des formes modulaires de Hilbert de poids demi-entier. La démonstration exploite aussi bien la représentation intégrales de Rankin-Selberg de la convolution que les formules explicites de Shimura [Duke Math. J., 52 (1985), 281-314] des coefficients de Fourier de certaines séries d’Eisenstein de poids demi-entier.

In this paper we construct p-adic measures related to the values of convolutions of Hilbert modular forms of integral and half-integral weight at the negative critical points under the assumption that the underlying totally real number field F has class number h F =1. This extends the result of Panchishkin [Lecture Notes in Math., 1471, Springer Verlag, 1991 ] who treated the case that both modular forms are of integral weight. In order to define the measures, we need to introduce the twist operator and a certain inverter j on the space of Hilbert modular forms of half-integral weight. The proof then makes use of the Rankin-Selberg integral representation of the convolution and of explicit formulas for the Fourier coefficients of certain Eisenstein series of half-integral weight derived by Shimura [Duke Math. J., 52 (1985), 281-314].

@article{AIF_1997__47_2_365_0,
     author = {D\"unger, Volker},
     title = {$p$-adic interpolation of convolutions of {Hilbert} modular forms},
     journal = {Annales de l'Institut Fourier},
     pages = {365--428},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {2},
     year = {1997},
     doi = {10.5802/aif.1569},
     zbl = {0882.11025},
     mrnumber = {98b:11050},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1569/}
}
TY  - JOUR
AU  - Dünger, Volker
TI  - $p$-adic interpolation of convolutions of Hilbert modular forms
JO  - Annales de l'Institut Fourier
PY  - 1997
SP  - 365
EP  - 428
VL  - 47
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1569/
DO  - 10.5802/aif.1569
LA  - en
ID  - AIF_1997__47_2_365_0
ER  - 
%0 Journal Article
%A Dünger, Volker
%T $p$-adic interpolation of convolutions of Hilbert modular forms
%J Annales de l'Institut Fourier
%D 1997
%P 365-428
%V 47
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1569/
%R 10.5802/aif.1569
%G en
%F AIF_1997__47_2_365_0
Dünger, Volker. $p$-adic interpolation of convolutions of Hilbert modular forms. Annales de l'Institut Fourier, Tome 47 (1997) no. 2, pp. 365-428. doi : 10.5802/aif.1569. https://aif.centre-mersenne.org/articles/10.5802/aif.1569/

[D] P. Deligne, Valeurs de fonctions L et périodes d'intégrales, in: Proc. of Symp. in Pure Math., 33 (1979), Part 2, 313-346. | MR | Zbl

[DR] P. Deligne, K.A. Ribet, Values of abelian L-functions at negative integers over totally real fields, Inventiones Math., 59 (1980), 227-286. | EuDML | MR | Zbl

[G] R. Greenberg, Motives, in: Proc. of Symp. in Pure Math., 55 (1994), Part 2, 193-223. | Zbl

[H] E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen, Chelsea, 1948. | Zbl

[Hi] H. Hida, On Λ-adic forms of half integral weight for SL2/Q, in: Number Theory Paris 1992-1993, editor S. David, LMSLN 215, 139-166 (1995), Cambridge Univ. Press. | Zbl

[I] J. Im, Special values of Dirichlet series attached to Hilbert modular forms, Am. J. Math., 113 (1991), 975-1017. | MR | Zbl

[K] N. Katz, p-adic L-functions for CM fields, Inventiones Math., 49 (1978), 199-297. | EuDML | MR | Zbl

[Kob] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Second Edition, GTM 97, Springer Verlag, 1993. | MR | Zbl

[Koh] W. Kohnen, Newforms of half-integral weight, J. Reine Angew. Math., 333 (1982), 32-72. | EuDML | MR | Zbl

[M] T. Miyake, Modular forms, Springer Verlag, 1989.

[MRV] M. Manickam, B. Ramakrishnan, T.C. Vasuvedan, On the theory of new-forms of half-integral weight, J. Number Th., 34 (1990), 210-224. | Zbl

[N] J. Neukirch, Algebraische Zahlentheorie, Springer Verlag, 1992. | Zbl

[P] A. Panchishkin, Non-archimedean L-functions of Siegel and Hilbert modular forms, Lecture Notes in Mathematics 1471, Springer Verlag, 1991. | MR | Zbl

[S1] G. Shimura, On modular forms of half-integral weight, Ann. of Math., 97 (1973), 440-481. | MR | Zbl

[S2] G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc., (3) 31 (1975), 79-98. | MR | Zbl

[S3] G. Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math., 29 (1976), 783-804. | MR | Zbl

[S4] G. Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J., 45 (1978), 637-679. | MR | Zbl

[S5] G. Shimura, Confluent hypergeometric functions on tube domains, Math. Ann., 260 (1982), 269-302. | MR | Zbl

[S6] G. Shimura, On Einsenstein Series of half-integral weight, Duke Math. J., 52 (1985), 281-314. | MR | Zbl

[S7] G. Shimura, On Hilbert modular forms of half-integral weight, Duke Math. J., 55 (1987), 765-838. | MR | Zbl

[S8] G. Shimura, On the Fourier coefficients of Hilbert modular forms of half-integral weight, Duke Math. J., 71 (1993), 501-557. | MR | Zbl

[U] M. Ueda, On twisting operators and newforms of half-integral weight, Nagoya Math. J., 131 (1993), 135-205. | MR | Zbl

Cité par Sources :