Galois representations attached to algebraic automorphic representations

This thesis is concerned with the Langlands program; namely the global Langlands correspondence, Langlands functoriality, and a conjecture of Gross. In chapter 1, we cover the most important background material needed for this thesis. This includes material on reductive groups and their root data, the definition of automorphic representations and a general overview of the Langlands program, and Gross' conjecture concerning attaching l-adic Galois representations to automorphic representations on certain reductive groups G over ℚ. In chapter 2, we show that odd-dimensional definite unitary groups satisfy the hypotheses of Gross' conjecture and verify the conjecture in this case using known constructions of automorphic l-adic Galois representations. We do this by verifying a specific case of a generalisation of Gross' conjecture; one should still get l-adic Galois representations if one removes one of his hypotheses but with the cost that their image lies in CG(ℚl) as opposed to LG(ℚl). Such Galois representations have been constructed for certain automorphic representations on G, a definite unitary group of arbitrary dimension, and there is a map CG(ℚl) → LG(ℚl) precisely when G is odd-dimensional. In chapter 3, which forms the main part of this thesis, we show that G = Un(B) where B is a rational definite quaternion algebra satisfies the hypotheses of Gross’ conjecture. We prove that one can transfer a cuspidal automorphic representation π of G to a π' on Sp2n (a Jacquet-Langlands type transfer) provided it is Steinberg at some finite place. We also prove this when B is indefinite. One can then transfer π′ to an automorphic representaion of GL2n+1 using the work of Arthur. Finally, one can attach l-adic Galois representations to these automorphic representations on GL2n+1, provided we assume π is regular algebraic if B is indefinite, and show that they have orthogonal image.