Siegel modular forms of genus $2$ attached to elliptic curves

Abstract

The object of this article is to construct certain classes of arithmetically significant, holomorphic Siegel cusp forms F of genus 2, which are neither of Saito-Kurokawa type, in which case the degree 4 spinor L-function L(s, F ) is divisible by an abelian L-function, nor of Yoshida type, in which case L(s, F ) is a product of L-series of a pair of elliptic cusp forms. In other words, for each F in the class of holomorphic Siegel cusp forms we construct below, which includes those of scalar weight ≥ 3, the cuspidal automorphic representation π of GSp(4,A) generated by F is neither CAP, short for Cuspidal Associated to a Parabolic, nor endoscopic, i.e., one arising as the transfer of a cusp form on GL(2)×GL(2), and consequently, L(s, F ) is not a product of L-series of smaller degrees. A key example of the first kind of Siegel modular forms we construct is furnished by the following result, which will be used in [34]: