Cuspidal Automorphic Representations Corresponding to Siegel Modular Forms

Abstract

In this chapter, we start with a cuspidal Hecke eigenform $F \in S_k(\Gamma _n)$ and construct an irreducible cuspidal automorphic representation of $\mathrm{GSp}_{2n}({\mathbb A})$ corresponding to it. There are several steps for achieving this— construct a function $\Phi _F$ on $\mathrm{GSp}_{2n}({\mathbb A})$ corresponding to F, understand the properties it inherits from F, and study the local components of the representation generated by $\Phi _F$. The main reference for this chapter is the article [6] by Asgari and Schmidt. We suggest the reader to go over Appendix B and C to refresh the details about adeles and local representation theory of $\mathrm{GL}_2$.