Local and Global Endoscopy for GSp(4)

Abstract

In this chapter we refine the global description of the endoscopic lift obtained in Corollary 4.2. The main result obtained in this chapter is Theorem 5.2, which is a special case of a global multiplicity formula conjectured by Arthur.We consider the symplectic group GSp(4) over an arbitrary totally real number field F. In the special case F = Q and for irreducible automorphic representations π with π_∞ in the discrete series the proof of this theorem is given in Sects. 5.2 and 5.3. In Sect. 5.1 we explain how the local endoscopic character lift is constructed for arbitrary local base fields of characteristic zero. In Sects. 5.4 and 5.5 we explain how the Arthur–Selberg trace formula can be used to extend the results obtained in Sects. 5.2 and 5.3 to the case of arbitrary totally real number fields and arbitrary irreducible automorphic not necessarily cohomological representations ?. The proof of Theorem 5.2 is based on two ingredients: the principle of exchange and the key formula (stated in Sect. 5.3). The latter can be directly deduced from weak versions of the trace formula (see Chap. 4, Corollary 4.2, or Sect. 5.5). It deals with simultaneous changes of global representations at two places. The principle of exchange, on the other hand, deals with an exchange of the representation at one place exclusively, and its proof boils down to a special case of the Hasse–Brauer–Noether theorem (Lemma 5.4). This is based on an explicit theta lift. At this point we use [56], which unfortunately forces us to make the restrictive assumption that F is a totally real number field. Since we apply results obtained in [56] in a rather specific case, it is very likely that this restriction on F can be removed.