Galois representations attached to algebraic automorphic representations
<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 g...
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| Format: | Thesis |
| Language: | English |
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2016
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| _version_ | 1826280000109674496 |
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| author | Green, B |
| author2 | Wiles, A |
| author_facet | Wiles, A Green, B |
| author_sort | Green, B |
| collection | OXFORD |
| description | <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> |
| first_indexed | 2024-03-07T00:07:12Z |
| format | Thesis |
| id | oxford-uuid:77f01cbc-65d1-480d-ae3a-0a039a76671a |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T00:07:12Z |
| publishDate | 2016 |
| record_format | dspace |
| spelling | oxford-uuid:77f01cbc-65d1-480d-ae3a-0a039a76671a2022-03-26T20:27:30ZGalois representations attached to algebraic automorphic representationsThesishttp://purl.org/coar/resource_type/c_db06uuid:77f01cbc-65d1-480d-ae3a-0a039a76671aMathematicsNumber theoryEnglishORA Deposit2016Green, BWiles, AWiles, A<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> |
| spellingShingle | Mathematics Number theory Green, B Galois representations attached to algebraic automorphic representations |
| title | Galois representations attached to algebraic automorphic representations |
| title_full | Galois representations attached to algebraic automorphic representations |
| title_fullStr | Galois representations attached to algebraic automorphic representations |
| title_full_unstemmed | Galois representations attached to algebraic automorphic representations |
| title_short | Galois representations attached to algebraic automorphic representations |
| title_sort | galois representations attached to algebraic automorphic representations |
| topic | Mathematics Number theory |
| work_keys_str_mv | AT greenb galoisrepresentationsattachedtoalgebraicautomorphicrepresentations |