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

Abstract

We state and verify up to weight 172 a conjecture on the existence of a certain generating set for spaces of classical Siegel modular forms. This conjecture is particularly useful for calculations involving Fourier expansions. Using this generating set, we verify the Böcherer conjecture for nonrational eigenforms and discriminants with class number greater than one. As a further application we verify another conjecture for weights up to 150 and investigate an analog of the Victor-Miller basis. Additionally, we describe some arithmetic properties of the basis we found.2010 Mathematics subject classification: primary 11F46; secondary 11F67, 11Y99.

Hyper-Algebras of Vector-Valued Modular Forms

“…We start with a conjecture, that we will not prove in this paper. There is experimental evidence in [15] that…”

Hecke eigenvalues of Siegel modular forms of “different weights”

“…The structure of the spaces of scalar-valued Siegel modular forms of degree 2 and level 1 is well-known thanks to results of Igusa; these results, together with the interplay between Jacobi modular forms and Siegel modular forms, have allowed the explicit decomposition of these spaces into eigenspaces for the Hecke operators. This approach was introduced by Skoruppa [22] and exploited and refined by a number of authors, most recently Raum [19]. Bases for these spaces are computed as sets of explicit Fourier expansions, and the effect of the operators T (p), T j (p 2 ) on these Fourier expansions can then be computed using the explicit formulas from [3] or [14].…”

“…The structure of the spaces of scalar-valued Siegel modular forms of degree 2 and level 1 is well-known thanks to results of Igusa; these results, together with the interplay between Jacobi modular forms and Siegel modular forms, have allowed the explicit decomposition of these spaces into eigenspaces for the Hecke operators. This approach was introduced by Skoruppa [22] and exploited and refined by a number of authors, most recently Raum [19].…”

Formulas for central values of twisted spin $L$-functions attached to paramodular forms

“…In Böcherer's original paper [Böc86] it was proved for F that are Saito-Kurokawa lifts and later Böcherer and Schulze-Pillot [BSP92] proved the conjecture in the case when F is a Yoshida lift. Kohnen and Kuss [KK02] gave numerical evidence in the case when F is of level one, degree 2 and is not a Saito-Kurokawa lift (these computations have recently been extended by Raum [Rau10]). A much more general approach to the conjecture has been pursued by Furusawa, Martin and Shalika [Fur93,FM11,FS99,FS00,FS03].…”