p-adic interpolation of convolutions of Hilbert modular forms
Annales de l'Institut Fourier, Volume 47 (1997) no. 2, p. 365-428
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].
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.
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {47},
     number = {2},
     year = {1997},
     pages = {365-428},
     doi = {10.5802/aif.1569},
     zbl = {0882.11025},
     mrnumber = {98b:11050},
     language = {en},
     url = {http://www.numdam.org/item/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/item/AIF_1997__47_2_365_0/

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