Proofs of the main theorems

Abstract

This long chapter is the technical heart of the book. We first apply Arthur’s theory to PGSp 4 ≃SO 3,2 to study the standard parameter of the 4 vector-valued genus 2 Siegel modular forms of interest for Niemeier lattices. Then, we apply Arthur’s theory to give two short, but conditional, proofs of Theorem E of the introduction, as well as a full proof that Theorem F implies Theorem E. We prove Theorem F by an in depth study of the Weil explicit formula applied to the L-functions of pairs of cuspidal algebraic automorphic representations of general linear groups. We then use Theorem F to classify cuspidal Siegel modular forms of weight ≤ 12 (Theorems D and G of the introduction).