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Aspects of Automorphic Induction

Belfanti, Edward Michael, Jr.
http://rave.ohiolink.edu/etdc/view?acc_num=osu1525706818378677

Abstract Details

2018, Doctor of Philosophy, Ohio State University, Mathematics.
Langlands' functoriality conjectures predict how automorphic representations of different groups are related to one another. Automorphic induction is a basic case of functoriality motivated by Galois theory. Let F be a local field of characteristic 0. The local Langlands correspondence for GL(N) states that there is a bijection between N-dimensional, complex representations of the Weil-Deligne group W_F' and irreducible, admissible representations of GL(N,F). Given an operation on Weil group representations, one can ask what the corresponding operation is for representations of GL(N,F). Automorphic induction is the operation on representations of GL(N,F) corresponding to induction of representations of the Weil group. Given a cyclic extension E of F of degree D, automorphic induction is a mapping of representations of GL(M,E) to representations of GL(MD,F). Once automorphic induction has been established for local fields, one can ask if the operation applied at each local component of an automorphic representation produces another automorphic representation. The automorphic induction problem was first considered in detail by Kazhdan. Kazhdan proved the local automorphic induction map exists in the of M=1. The next major result is due to Arthur-Clozel, where it was shown that a global automorphic induction operation exists for prime degree extensions. The local theory was completed by Henniart and Herb; they showed that the local automorphic induction map exists for arbitrary M and cyclic extensions of arbitrary degree. Moreover, Henniart showed later that the local automorphic induction map is consistent with induction of Weil group representations and the local Langlands correspondence. Finally, Henniart extended the results of Arthur-Clozel to cyclic extensions of any degree and verified that the resulting mapping of automorphic representations is consistent with the local lifting. All of these results rely on some version of the trace formula and powerful theorems about L-functions for GL(N). The goal of this thesis is to give a different proof of local and global automorphic induction when M=1 and to emphasize some different aspects of the theory. As with the results previously mentioned, the main technical tool is a trace formula, specifically the full global trace formula of Arthur. The proof relies more on the trace formula and less on L-functions than previous proofs. Chapter 1 consists of background and preliminaries needed to state the main theorems on local and global automorphic induction. We also discuss various results that are needed to state Arthur's trace formula. In Chapter 2, we give a statement of the relevant trace formulas. We also discuss some computations in general rank that could be used in a generalization of our techniques to the case of M>1. The main local and global theorems are established in Chapter 3. As with most trace formula arguments, the proof boils down to an application of linear independence of characters.
James Cogdell (Advisor)
233 p.
Mathematics
Automorphic representations, trace formulas

Recommended Citations

Citations

  • Belfanti, Jr., E. M. (2018). Aspects of Automorphic Induction [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1525706818378677

    APA Style (7th edition)

  • Belfanti, Jr., Edward. Aspects of Automorphic Induction. 2018. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1525706818378677.

    MLA Style (8th edition)

  • Belfanti, Jr., Edward. "Aspects of Automorphic Induction." Doctoral dissertation, Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu1525706818378677

    Chicago Manual of Style (17th edition)

osu1525706818378677
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This open access ETD is published by The Ohio State University and OhioLINK.