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 y2 = x5 - 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.

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