L-functions of $S_3(\G_2(2,4,8))$

TL;DR

This paper investigates the L-functions of Siegel cuspforms of degree 2 related to a specific congruence subgroup, confirming some conjectures, classifying automorphic representations, and constructing new modular forms and differential forms.

Contribution

It proves the explicit description of certain L-functions, classifies automorphic representations of O(6), and constructs new Hermitian and differential forms related to the subgroup.

Findings

Confirmed conjectures on Andrianov L-functions for these cuspforms.

Identified automorphic representations of O(6) associated with the forms.

Constructed non-holomorphic differential three-forms on the Siegel threefold.

Abstract

The space of Siegel cuspforms of degree $2$ of weight $3$ with respect to the congruence subgroup $\G_2(2,4,8)$ was studied by van Geemen and van Straten in Math. computation. {\bf 61} (1993). They showed the space is generated by six-tuple products of Igusa $\th$-constants, and all of them are Hecke eigenforms. They gave conjecture on the explicit description of the Andrianov $L$-functions. In J. Number Theory. {\bf 125} (2007), we proved some conjectures by showing that some products are obtained by the Yoshida lift, a construction of Siegel eigenforms. But, other products are not obtained by the Yoshida lift, and our technique did not work. In this paper, we give proof for such products. As a consequence, we determine automorphic representations of O(6), and give Hermitian modular forms of SU(2,2) of weight $4$. Further, we give non-holomorphic differential threeforms on the Siegel threefold with respect to $\G_2(2,4,8)$.