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Fields of Rationality of Cuspidal Automorphic Representations MASSACHUSETTS INSTITUTE OF TECHNOLOGY by JUN John Binder 1E6 201 LIBRARIES Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of ARCHIES Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2016 @ Massachusetts Institute of Technology 2016. All rights reserved. A Signature redacted A uthor ................................ ......... ..y... ................ Department of Mathematics April 7, 2016 ignature redacted Certified by..... S Sug Woo Shin Associate Professor of Mathematics, University of California, Berkeley Thesis Supervisor Accepted by ...... Signature redacted Alexei Borodin Chairman, Mathematics Department Chair of Graduate Studies 2 Fields of Rationality of Cuspidal Automorphic Representations by John Binder Submitted to the Department of Mathematics on April 7, 2016, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract This thesis examines questions related to the growth of fields of rationality of cuspidal automorphic representations in families. Specifically, if F is a family of cuspidal automorphic representations with fixed central character, prescribed behavior at the Archimedean places, and such that the finite component 7r' has a F-fixed vector, we expect the proportion of 7r E F with bounded field of rationality to be close to zero if F is small enough. This question was first asked, and proved partially, by Serre for families of classical cusp forms of increasing level. In this thesis, we will answer Serre's question affirmatively by converting the question to a question about fields of rationality in families of cuspidal automorphic GL2 (A) representations. We will consider the analogous question for certain sequences of open compact subgroups F in UE/F(n). A key intermediate result is an equidistribution theorem for the local components of families of cuspidal automorphic representations. Thesis Supervisor: Sug Woo Shin Title: Associate Professor of Mathematics, University of California, Berkeley 3 4 Acknowledgments A great many people have helped, directly or indirectly, in the preparation of this thesis. I am grateful to all of them, and I regret that I can't possibly list everybody who has touched my life during this process. First and foremost, my family provided me unfailing support over the course of graduate school. Mom, Dad, Kevin, and Kristin: thank you so much for always being there to provide an ear or some levity during these five years. I couldn't have done it without you guys, and I love you all. My friends, both within and outside the mathematics department, have been a source of strength. Thanks to Dan Tureza, Sean Simmons, and Hans Liu for putting up with me as a roommate for 4, 4, and 2 years, respectively. Thanks to Mark Rosenberg for his unwavering friendship and for always finding time to pick up the phone just to chat, whether about the tribulations of graduate school, fantasy football, or any subject in between. The list of MIT classmates who have touched my life is too long to name exhaustively, but I am grateful to each of them. I'd specifically like to mention Ruthi Hortsch and Padma Srinivasan for keeping me sane during the long hours we spent finishing algebraic geometry and number theory homework and for remaining true friends as we've worked through the PhD process. My 'little brother,' Adrian, has been a constant joy since he and I were matched in January, 2013. I'm grateful to him for keeping me young at heart, and to his family for inviting me into their home and their circle. The MIT Frisbee team provided a much needed release from the day-to-day grind of research. I'll always cherish my years as a member of the team and as an assistant coach. The MIT math department staff has been a wonderful source of support and friendship, and I can't begin to list everything they deserve credit for. Some high- lights include: getting the department from and to building 2 during our 3-year-long renovation process, keeping me company whenever I came in for a study break, an- swering my myriad questions about thesis requirements and, of course, allowing me 5 to come in and play with Rocco multiple times a week. Thank you all for everything! Many professors have contributed to my mathematical development, both as an undergraduate at the University of Chicago and as a graduate student at MIT. I'd specifically like to mention Bjorn Poonen, who first called me to offer me admittance to MIT, who has been a constant presence in the number theory group, and who was the first professor to teach me about schemes. Abhinav Kumar and Gigliola Staffilani served on my qualifying exam committee and were incredibly supportive through an inherently stressful process. It has been wonderful to have Julee Kim and David Vogan on my thesis committee. Thank you both for the time you've spent reading my and critiquing my work, and for taking the time to explain the finer points of p-adic and real representation theory, respectively. Most importantly, I am grateful to my adviser, Sug Woo Shin, for his consistent mentorship, friendship, and support. First, and foremost, thank you for suggesting this thesis topic and for allowing me the freedom to take the problem in directions that interested me. Thank you for correcting my many misunderstandings, for helping me flesh out numerous half-baked ideas, and finally for allowing me to escape the cold and visit Berkeley twice a year. I'm honored to be your first PhD student. This thesis is dedicated to the memory of Paul Sally, who first taught me how to compute Fourier transforms on Qp, and who took time out of his busy schedule to explain Tate's Thesis to me when I was a wide-eyed college student. 6 This doctoral thesis has been examined by a Committee of the Department of Mathematics as follows: Signature redacted Professor Julee Kim .... Chairman, Thesis Committee Professor of Mathematics Signature redacted Professor Sug Woo Shin ..... .0 I, Thesis Supervisor Associate Professor of Mathematics, University of California, Berkeley Signature redacted Professor David Vogan.. ......... Il4ember, Thesis Committee Professor of Mathematics 8 Contents 1 Introduction 11 1.1 History and motivation .. ........................ 11 1.2 Families of automorphic representations .. ...... ...... ... 13 1.3 Outline of the paper ...... ............. ........ 15 2 Basics on Representation Theory and Unitary Groups 19 2.1 Admissible representations of p-adic groups .... ........... 19 2.2 The tempered spectrum of GL,(L) .. ............... ... 22 2.2.1 Generic representations and conductors ..... ........ 25 2.3 The global situation: discrete and cuspidal automorphic representations 26 2.4 The groups UE/F(n); maximal special subgroups .. ......... 28 3 Plancherel Equidsitribution 31 3.1 Hecke algebras and Plancherel measure . ............. ... 31 3.2 Euler-Poincar6 functions at the Archimedean places .. ........ 34 3.3 Counting measures and Plancherel equidistribution .... ...... 36 3.4 The trace formula and density theorem ...... ...... ..... 38 3.5 Plancherel equidistribution for more general ..... ........ 43 4 Bounds on trace formula terms 47 4.1 Bounds on trace formula terms for Fo(pr) < GL2 (Fp) ... ...... 47 4.1.1 Computation of constant terms ....... .......... 50 4.1.2 Computation of orbital integrals ........ ......... 52 9 4.1.3 Summary of global consequences ...... .......... 54 4.2 General bounds for trace formula terms on U(n) ...... ...... 55 5 Refined Plancherel equidistribution for varying conductor 59 5.1 Definition of enew ........................ ..... 60 5.2 Analysis of trace formula terms and Plancherel equidistribution ... 63 6 Basics on fields of rationality; local representations 69 6.1 Fields of rationality of supercuspidal representations in the tame case 71 6.2 Fields of rationality of discrete series and tempered representations . 82 7 Statements and Proofs of the Main Theorems 85 7.1 Fields of rationality in families: statements of the main theorems . .. 85 7.2 A finiteness result for local components of cohomological representations 89 7.3 Contingent completion of the proof ........ ........ ... 92 8 Properties of depth-zero discrete series representations 99 8.1 Counting discrete series representations with small field of rationality 101 8.2 Proof of 8.0.0.4 in some cases ........... ........... 103 8.3 Hecke algebra isomorphisms ..... ............ ...... 109 8.4 The local character expansion ...... ........ ........ 111 10 Chapter 1 Introduction 1.1 History and motivation Let Sk(Fo(N)) denote the space of classical cusp forms of weight k and level N. For each prime p f N, the Hecke operator Tp is semisimple on Sk(Fo(N)). Since the Tp commute with one another, they are simultaneously diagonalizable. By Atkin-Lehner theory ([AL70]), there is in fact a canonical eigenbasis for the space, which we denote by Bk(Fo(N)). If f E Bk(Fo(N)) then f has a Fourier expansion f(q) = q + a2q2 + a3 q .... We define the field of rationality Q(f) of f to be the field Q(a 2 , a3 , ... ) generated by all its Fourier coefficients. If o- E Aut(C), then the form 2 3 of(q) = q+ o-(a 2)q + (a3)q +... is also a member of the canonical basis of Hecke eigenforms. Since this space is finite- dimensional, we discern that Q(f) is a finite extension of Q. Let BkA(Fo(N)) be the set of those f E Bk(Fo(N)) such that [Q(f) Q] < A. In [Ser971, Serre examined the growth of Q(f) in level-families of Hecke eigenforms. 11 Specifically, he proved the following Theorem: Theorem 1.1.0.1. Fix A > 1 and a rational prime po. Let N, -+ oc be a sequence of levels such that (po, NA) = 1 for all N\. Then lim JkmBA(Fo(NA))J \)I= 0. A-+00 IBk(1o(NA))I On page 87 of that paper, Serre posited it was possible to allow {NA} to be an arbitrary sequence of levels, removing the necessity of the auxiliary prime.

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