In this paper we construct -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 has class number . 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 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].
Dans cet article nous construisons des mesures -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 ait un nombre de classes . 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 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.
@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}, mrnumber = {98b:11050}, zbl = {0882.11025}, language = {en}, url = {http://www.numdam.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 - http://www.numdam.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 http://www.numdam.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, Volume 47 (1997) no. 2, pp. 365-428. doi : 10.5802/aif.1569. http://www.numdam.org/articles/10.5802/aif.1569/
[D] Valeurs de fonctions L et périodes d'intégrales, in: Proc. of Symp. in Pure Math., 33 (1979), Part 2, 313-346. | MR | Zbl
,[DR] Values of abelian L-functions at negative integers over totally real fields, Inventiones Math., 59 (1980), 227-286. | EuDML | MR | Zbl
, ,[G] Motives, in: Proc. of Symp. in Pure Math., 55 (1994), Part 2, 193-223. | Zbl
,[H] Vorlesungen über die Theorie der algebraischen Zahlen, Chelsea, 1948. | Zbl
,[Hi] 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] Special values of Dirichlet series attached to Hilbert modular forms, Am. J. Math., 113 (1991), 975-1017. | MR | Zbl
,[K] p-adic L-functions for CM fields, Inventiones Math., 49 (1978), 199-297. | EuDML | MR | Zbl
,[Kob] Introduction to Elliptic Curves and Modular Forms, Second Edition, GTM 97, Springer Verlag, 1993. | MR | Zbl
,[Koh] Newforms of half-integral weight, J. Reine Angew. Math., 333 (1982), 32-72. | EuDML | MR | Zbl
,[M] Modular forms, Springer Verlag, 1989.
,[MRV] On the theory of new-forms of half-integral weight, J. Number Th., 34 (1990), 210-224. | Zbl
, , ,[N] Algebraische Zahlentheorie, Springer Verlag, 1992. | Zbl
,[P] Non-archimedean L-functions of Siegel and Hilbert modular forms, Lecture Notes in Mathematics 1471, Springer Verlag, 1991. | MR | Zbl
,[S1] On modular forms of half-integral weight, Ann. of Math., 97 (1973), 440-481. | MR | Zbl
,[S2] On the holomorphy of certain Dirichlet series, Proc. London Math. Soc., (3) 31 (1975), 79-98. | MR | Zbl
,[S3] The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math., 29 (1976), 783-804. | MR | Zbl
,[S4] The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J., 45 (1978), 637-679. | MR | Zbl
,[S5] Confluent hypergeometric functions on tube domains, Math. Ann., 260 (1982), 269-302. | MR | Zbl
,[S6] On Einsenstein Series of half-integral weight, Duke Math. J., 52 (1985), 281-314. | MR | Zbl
,[S7] On Hilbert modular forms of half-integral weight, Duke Math. J., 55 (1987), 765-838. | MR | Zbl
,[S8] On the Fourier coefficients of Hilbert modular forms of half-integral weight, Duke Math. J., 71 (1993), 501-557. | MR | Zbl
,[U] On twisting operators and newforms of half-integral weight, Nagoya Math. J., 131 (1993), 135-205. | MR | Zbl
,Cited by Sources: