Surjectivity of Siegel $\Phi$-operator for square free level and small weight

Böcherer, Siegfried; Ibukiyama, Tomoyoshi

Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 121-144.

Nous démontrons la surjectivité de l’opérateur $\Phi$ de Siegel pour des formes modulaires pour certains groupes de congruence de $\mathrm{Sp}(2,\mathbb{Z})$ et de poids 4, où les techniques standards (séries de Poincaré ou séries de Klingen-Eisenstein) ne marchent pas. Nous utilisons des séries thêta et le problème de base pour plusieurs genres.

We show the surjectivity of the (global) Siegel $\Phi$-operator for modular forms for certain congruence subgroups of $\mathrm{Sp}(2,\mathbb{Z})$ and weight $k=4$, where the standard techniques (Poincaré series or Klingen-Eisenstein series) are no longer available. Our main tools are theta series and genus versions of basis problems.

Classification : 11F46, 11F27
Keywords: Siegel modular form, $\Phi $-operator, Theta series
Mot clés : formes modulaires de Siegel, l’opérateur $\Phi $, séries de thêta

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