L-functions of S₃(₂(2,4,8))

الأصل · EN

The space of Siegel cuspforms of degree 2 of weight 3 with respect to the congruence subgroup ₂(2,4,8) was studied by van Geemen and van Straten in Math. computation. 61 (1993). They showed the space is generated by six-tuple products of Igusa þ-constants, and all of them are Hecke eigenforms. They gave conjecture on the explicit description of the Andrianov L-functions. In J. Number Theory. 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 ₂(2,4,8).