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Aspects of Automorphic Induction DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Edward Michael Belfanti, Jr., B.A. Graduate Program in Mathematics The Ohio State University 2018 Dissertation Committee: James W. Cogdell, Advisor Sachin Gautam Roman Holowinsky c Copyright by Edward Michael Belfanti, Jr. 2018 Abstract 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 0 representations of the Weil-Deligne group WF 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 ⊃ 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 in [Kaz84]. Kazhdan proved the local automorphic induction map exists in the of M = 1. The next major result was [AC89], where it was shown that a global automorphic induction operation exists for prime degree extensions. The local theory was completed in [HH95]; Henniart and Herb showed that the local automorphic induction map exists for arbitrary M and cyclic extensions of arbitrary degree. Moreover, Henniart showed in [Hen01] that the local automorphic induction map constructed in [HH95] is consistent with induction of Weil group representations and the local Langlands correspondence. Finally, Henniart extended the results of [AC89] in [Hen12] to cyclic extensions of any degree and verified that the resulting ii mapping of automorphic representations is consistent with the local lifting of [HH95]. 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. iii For my family, friends, and all those who helped along the way iv Acknowledgments A dissertation is the work of many people, some of whom deserve special thanks. Melissa Grasso has been with me through not one, but two final semesters during which I was convinced I would never graduate. Her love, support and companionship over the past seven years, and through some particularly trying months, has helped me more than I can ever repay. Her own successes and diligent hard work provided the motivation I needed to see this project through to the end; I only hope that the completion of this degree justifies in some small way the innumerable sacrifices, big and small, she has made for me. With love, thank you. My parents, Ed and Teresa, have provided boundless love, generosity and advice, not just these past seven years but the past twenty-nine. I feel fortunate every day for their support, and it is an honor to be their son and share any success I may find with them. Thank you always. My brothers, Nick and Dom, have provided the support and encouragement that their older brother is supposed to provide. Thank you both for never doubting me. Thanks to Nana, Papa, and Grandma, for the love and support unique to grandparents. Thanks to my Grandpa, who I know would be proud of me. Thanks to my aunts and uncles Amy, Paul and Mary, and my cousins William, Charles and Kate, for a fun Christmas before getting back to the grind. I've received many kinds words from my aunts Nancy, Mary, Donna and Julie, which were especially welcome over the last weekend of work on this document. I would like to thank Linda James, John Grasso, Jennifer Grasso, Erika James and Kevin Riopelle, for always making me feel welcome in Sudbury. v Stephen Eigenmann deserves thanks for always helping me keep my priorities straight. Thanks to Jack Marczak, with whom I started studying math in high school. Thanks also to Larry Marczak, without whose help I may never have mastered logarithms. I would like to thank the group chat consisting of Matt Flagg, Eric Kim, Rob Monaco and Tomas Castella for providing needed humor over the past seven years and for keeping me apprised of the news and scores I missed while completing this work. Matt deserves special thanks for his commiseration and support throughout more than ten years of school. John Lynch and I have been friends ever since we realized we were in so many classes together that it was getting weird for us not to talk. Carney forever. I would like to thank my friends and former roommates Steph Wilker, Scott Jaffee and Matt Jaffee, who managed to tolerate me through my early years in grad school, and qualifying exams in particular. Thanks to Catherine Glover and Lyman Gillispie for their friendship, and for getting me out of my office. A special thanks to Lyman, for always taking seriously my terrible opinions about books that I don't understand. I would like to thank my colleague Yilong Wang for interesting conversations, mathe- matical and otherwise, and for encouragement over the last few weeks of writing. Dan Moore, PhD, has been a close friend ever since he got over my gum snapping. His technical support kept me sane at a very strenuous time during the completion of this document. Thanks for the many, let's say bizarre and enlightening, conversations. We'll always have that Culver's on the outskirts of middle of nowhere Indiana. Dan could not have supported me without the support he received from Mariel Colman, who deserves special thanks for her patience with us. Thanks to Solomon Friedberg, whose advice and encouragement when I was at Boston College inspired me to go to graduate school, and study representation theory in particular. Jim Cogdell, my advisor, deserves thanks for many hours of conversation about repre- sentation theory, as well as for frantically reading this thesis and providing feedback as I pushed every conceivable deadline. It is impossible to believe this document could exist without his wisdom and advice throughout the research and writing process. He deserves vi special thanks for helping me through a difficult project and situation not of his design. vii Vita 2007{2011 . Boston College B.A. in Mathematics 2011{Present . The Ohio State University Graduate Teaching Associate, Graduate Research Associate Fields of Study Major Field: Mathematics Studies in Automorphic Representation Theory: James W. Cogdell viii Table of Contents Page Abstract........................................... ii Dedication......................................... iv Acknowledgments.....................................v Vita............................................. viii Chapters 1 Preliminaries ..................................... 1 1.1 L-groups...................................... 1 1.2 Twisted endoscopy for (GN ;!) ......................... 11 1.3 Representations of GN .............................. 22 1.4 A substitute for global parameters ....................... 44 1.5 Langlands-Shelstad-Kottwitz transfer...................... 64 1.6 Local intertwining operators........................... 79 1.7 Statement of the main theorems ........................ 99 2 Trace formulas .................................... 108 2.1 The discrete part of the trace formula ..................... 108 2.2 Stabilization.................................... 114 2.3 Contribution of a parameter . 121 2.4 A preliminary comparison............................ 128 2.5 The stable multiplicity formula......................... 135 2.6 An elliptic computation ............................. 146 2.7 Globalizing local representations ........................ 153 2.8 Orthogonality relations.............................. 164 3 The case of M = 1.................................. 170 3.1 Stable multiplicity when M = 1......................... 170 3.2 Orthogonality relations when M = 1...................... 171 3.3 Spherical characters and weak lifting...................... 173 3.4 Main theorems .................................. 178 Bibliography ....................................... 191 ix Appendices A Odds and ends ...................................
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