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 <sup>C</sup>G(ℚ<sub>l</sub>) as opposed to <sup>L</sup>G(ℚ<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>G(ℚ<sub>l</sub>) → <sup>L</sup>G(ℚ<sub>l</sub>) precisely when G is odd-dimensional. In chapter 3, which forms the main part of this thesis, we show that G = U<sub>n</sub>(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 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 πâ² 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 Ï is regular algebraic if B is indefinite, and show that they have orthogonal image.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:730056 |
| Date | January 2016 |
| Creators | Green, Benjamin |
| Contributors | Wiles, Andrew |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://ora.ox.ac.uk/objects/uuid:77f01cbc-65d1-480d-ae3a-0a039a76671a |
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