| Summary: | <p>This thesis is concerned with the Langlands program; namely the global Langlands correspondence, Langlands functoriality, and a conjecture of Gross.</p> <p>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 <em>l</em>-adic Galois representations to automorphic representations on certain reductive groups <em>G</em> over &Qopf;.</p> <p>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 <em>l</em>-adic Galois representations. We do this by verifying a specific case of a generalisation of Gross' conjecture; one should still get <em>l</em>-adic Galois representations if one removes one of his hypotheses but with the cost that their image lies in <sup>C</sup><em>G</em>(&Qopf;<sub>l</sub>) as opposed to <sup>L</sup><em>G</em>(&Qopf;<sub>l</sub>). Such Galois representations have been constructed for certain automorphic representations on G, a definite unitary group of arbitrary dimension, and there is a map <sup>C</sup><em>G</em>(&Qopf;<sub>l</sub>) → <sup>L</sup><em>G</em>(&Qopf;<sub>l</sub>) precisely when <em>G</em> is odd-dimensional.</p> <p>In chapter 3, which forms the main part of this thesis, we show that <em>G</em> = U<sub>n</sub>(B) where <em>B</em> is a rational definite quaternion algebra satisfies the hypotheses of Gross’ conjecture. We prove that one can transfer a cuspidal automorphic representation π of <em>G</em> to a π' on Sp<sub>2n</sub> (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 GL<sub>2n+1</sub> using the work of Arthur. Finally, one can attach l-adic Galois representations to these automorphic representations on GL<sub>2n+1</sub>, provided we assume π is regular algebraic if <em>B</em> is indefinite, and show that they have orthogonal image.</p>
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