Character identities and Galois representations related to the group GSp(4)

Abstract

In this chapter we study the l–adic representations attached to cohomological Siegel modular forms. To understand their associated l–adic representations defined by the Galois action on the etale cohomology, one needs to understand the endoscopic lifts and vice versa, since for all automorphic representations in this lift the associated l–adic representations turn out to be smaller than what would be expected a priori.

The simple version of the Lefschetz trace formula used in Chap. 3 suffers from the particular restriction that for one specific prime – the chosen Frobenius prime p – the corresponding local Hecke operators at pp always have to be deleted from consideration. As a consequence, this particular chosen Frobenius prime p together with the Archimedean place play a distinguished role. This situation in fact looks similar to the situation in Arthur’s simple trace formula. Of course this is not a mere coincidence, but is definitely forced by the failure of the strong multiplicity 1 theorem for automorphic forms on GSp(4). To bypass the technical difficulties that result from this, the l–adic representations of the absolute Galois group on the cohomology turn out to be very helpful. With their help we analyze the endoscopic lift. This allows us to understand the failure of the strong multiplicity 1 theorem caused by it.