Cusp forms of weight 2 for the group \(\Gamma_ 2(4,8)\). (English) Zbl 0780.11024

Salvati-Manni, Riccardo; Top, Jaap

Am. J. Math. 115, No. 2, 455-486 (1993).

Stimulated by such renowned conjectures like the conjecture of Ramanujan or that of Taniyama-Weil which are linked to cusp forms of weight 2 to certain congruence subgroups of the classical elliptic modular group the authors aim at providing examples of cusp forms of weight 2 for congruence subgroups of \(\text{Sp}_ 4(\mathbb Z)\). Starting with Igusa’s generators they succeed in doing this for the group \(\Gamma_ 2(4,8)\): the space of all modular forms of weight 2 is 695-dimensional whereas the subspace of cusp forms has dimension 15 and is spanned by 15 products of four theta constants \(\Theta_ M\) respectively. This is being done by decomposing the whole space in a direct sum of character spaces for the action of \(\Gamma_ 2(2)\). Moreover these products of theta constants are all eigenfunctions of the Hecke operators \(T_ 2(m)\) for odd \(m\) with eigenvalues \(\lambda(m)=0\) and \(\lambda(m^ 2)=m^ 2\) in the case \(m\) prime \(\equiv 3\pmod 4\), thus giving a counterexample to the generalized Ramanujan conjecture.
As an immediate consequence all these products have \(\zeta_{\mathbb Q(i)} (s)\zeta_{\mathbb Q(i)}(s-1)\) as Andrianov \(L\)-function.
Finally miscellaneous results for some other subgroups of \(\text{Sp}_ 4(\mathbb Z)\) are gathered.

Reviewer: F. W. Knoeller (Marburg)

MSC:

11F46	Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F60	Hecke-Petersson operators, differential operators (several variables)

Keywords:

cusp forms of weight 2; congruence subgroups of \(\text{Sp}_ 4(\mathbb Z)\); theta constants; Hecke operators; Andrianov \(L\)-function