A Böcherer-Type Conjecture for Paramodular Forms

Abstract

In the 1980s Böcherer formulated a conjecture relating the central value of the quadratic twists of the spinor L-function attached to a Siegel modular form F to the coefficients of F . He proved the conjecture when F is a Saito-Kurokawa lift. Later Kohnen and Kuß gave numerical evidence for the conjecture in the case when F is a rational eigenform that is not a Saito-Kurokawa lift. In this paper we develop a conjecture relating the central value of the quadratic twists of the spinor L-function attached to a paramodular form and the coefficients of the form. We prove the conjecture in the case when the form is a Gritsenko lift and provide numerical evidence when it is not a lift.

Efficiently Generated Spaces of Classical Siegel Modular Forms and the Böcherer Conjecture

“…The effort made to explore paramodular forms of low weight in [13] should also be mentioned. Building on this and the paramodular conjecture (see [5]), some spinor L-series attached to paramodular forms were calculated in [16]. This last work is particularly significant since it begins to build the picture for high-level paramodular groups.…”

“…The case of Yoshida lifts of weight two has also been treated (see [3]). The author [14] and Ryan and Tornaría [16] recently argued that the conjecture should also hold in the paramodular setting, and proved that it holds for Gritsenko lifts. For [3] Spaces of classical Siegel modular forms 395…”

The Maass Space for Paramodular Groups

“…Recently paramodular forms of degree two and squarefree level N came into focus again. For example • The paramodular conjecture by Brumer and Kramer [4] • New and old form theory by Roberts and Schmidt [18] • The Böcherer conjecture for paramodular forms [19] • Borcherds lifts [3] Most of the results have in common that a certain subspace of generalized Saito-Kurokawa lifts, the Maaß space, plays a significant role. Although several authors have studied these lifts in the orthogonal setting (cf.…”