PROOF OF R. SALVATI MANNI AND J. TOP’S CONJECTURES ON SIEGEL MODULAR FORMS AND ABELIAN SURFACES

One of the conjectures claims that the Hasse-Weil zeta function corresponding to the Jacobian variety defined over Q 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. 1. Introduction and main idea. H. Yoshida conjectured in [Y1] that every Hasse-Weil zeta function of abelian surface over Q is equal to some Siegel modular form of degree 2 and weight 2 if w = 1, where w is the constant in the conjectural functional equation. (I heard that J. Shalika gave the same conjecture.) R. Salvati Manni and J. Top’s conjectures provide some candidates of suc