Proof of R. Salvati Manni and J. Top's conjectures on Siegel modular forms and Abelian surfaces

Abstract

This paper gives complete proofs of R. Salvati Manni and J. Top's conjectures which are contained in the paper, "Cusp forms of weight 2 for the group Γ(4, 8), Amer. J. Math . 115 (1993), 455-486." One of the conjectures claims that the Hasse-Weil zeta function corresponding to the Jacobian variety defined over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] of the hyper-elliptic curve y 2 = x 5 - x equals the Andrianov L -function of Siegel cusp form Θ of degree 2 and weight 2 which is four products of the Igusa theta constants. Regarding the Θ as a function obtained by the Yoshida lifting from a pair of elliptic modular forms, we prove the conjecture.