$p$-adic interpolation of convolutions of Hilbert modular forms

Dünger, Volker

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].

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