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.
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].
@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}, volume = {47}, year = {1997}, pages = {365-428}, doi = {10.5802/aif.1569}, mrnumber = {98b:11050}, zbl = {0882.11025}, language = {en}, url = {http://dml.mathdoc.fr/item/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) pp. 365-428. doi : 10.5802/aif.1569. http://gdmltest.u-ga.fr/item/AIF_1997__47_2_365_0/
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