On the modularity of the GL2-twisted spinor L-function

Abstract

In modern number theory there are famous theorems on the modular- ity of Dirichlet series attached to geometric or arithmetic objects. There is Hecke's converse theorem, Wiles proof of the Taniyama-Shimura conjecture, and Fermat's Last Theorem to name a few. In this article in the spirit of the Langlands philos- ophy we raise the question on the modularity of the GL2-twisted spinor L-function ZG h(s) related to automorphic forms G;h on the symplectic group GSp2 and GL2. This leads to several promising results and nally culminates into a pre- cise very general conjecture. This gives new insights into the Miyawaki conjecture on spinor L-functions of modular forms. We indicate how this topic is related to Ramakrishnan's work on the modularity of the Rankin-Selberg L-series