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Counterexamples Related to the Sato–Tate Conjecture

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Counterexamples Related to the Sato–Tate Conjecture COUNTEREXAMPLES RELATED TO THE SATO–TATE CONJECTURE A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Daniel Miller May 2017 c 2017 Daniel Miller ALL RIGHTS RESERVED COUNTEREXAMPLES RELATED TO THE SATO–TATE CONJECTURE Daniel Miller, Ph.D. Cornell University 2017 Let E/Q be an elliptic curve. The Sato–Tate conjecture, now a theorem, tells us a −1 pp that the angles qp = cos 2 p are equidistributed in [0, p] with respect to 2 2 1 1 the measure p sin q dq if E is non-CM (resp. 2p dq + 2 dp/2 if E is CM). In the non-CM case, Akiyama and Tanigawa conjecture that the discrepancy Z x 1 2 2 DN = sup ∑ 1[0,x](qp) − sin q dq p(N) 0 p x2[0,p] p6N − 1 +e asymptotically decays like N 2 , as is suggested by computational evidence and certain reasonable heuristics on the Kolmogorov–Smirnov statistic. This conjecture implies the Riemann hypothesis for all L-functions associated with E. It is natural to assume that the converse (“generalized Riemann hypothesis implies discrepancy estimate”) holds, as is suggested by analogy with Artin L-functions. We construct, for compact real tori, “fake Satake parameters” yielding L-functions which satisfy the generalized Riemann hypothesis, but for which the discrepancy decays like N−e for any fixed e > 0. This provides ev- idence that for CM abelian varieties, the converse to “Akiyama–Tanigawa con- jecture implies generalized Riemann hypothesis” does not follow in a straight- forward way from the standard analytic methods. We also show that there are Galois representations r : Gal(Q/Q) ! GL2(Zl), ramified at an arbitrarily thin (but still infinite) set of primes, whose Satake parameters can be made to converge at any specified rate to any fixed measure m on [0, p] for which cos∗ m is absolutely continuous with bounded derivative. BIOGRAPHICAL SKETCH Daniel Miller was born in St. Paul, Minnesota. He completed his Bachelor of Science at the University of Nebraska Omaha. In addition to his studies there, he played the piano competitively and attended Cornell’s Summer Mathemat- ics Institute. He started his Ph.D. at Cornell planning on a career in academia. Halfway through he had a change of heart, and will be joining Microsoft’s Anal- ysis and Experimentation team as a data scientist after graduation. He is hap- pily married to Ivy Lai Miller, and owns a cute but grumpy cat named Socrates. iii This thesis is dedicated to my undergraduate adviser, Griff Elder. He is the reason I considered a career in mathematics, and his infectious enthusiasm for number theory has inspired me more than I can say. iv ACKNOWLEDGEMENTS I could not have completed this thesis without help and support from many people. I would like to offer my sincerest thanks to the following people, and my sincerest apologies to anyone whose name I have forgotten to include here. My parents Jay and Cindy, for noticing and fostering my mathematical in- terests early on, and for being unfailingly loving and supportive. My undergraduate thesis advisor, Griffith Elder. Without his encouragement and inspiration I probably would have never considered a career in math. Tara Holm, Jason Boynton, and Anthony Weston, for making Cornell’s 2011 Summer Mathematics Institute the fantastic experience it was. My fellow graduate students Sasha Patotski, Balazs´ Elek, and Sergio Da Silva, for sharing my early love of algebraic geometry, laughing with me at the absurdities of academic life, and listening to my ramblings about number theory. The mathematics department at Cornell, where many professors were gen- erous with their time and ideas. I appreciate Yuri Berest, John Hubbard, Farbod Shokrieh, Birget Speh, and David Zywina for letting me bounce ideas off them, helping me add rigor to half-baked ideas, and pointing my research in new and interesting directions. My advisor Ravi Ramakrishna. He kindled my first love for number the- ory, stayed supportive as my research bounced all over the place, and kept me focused, grounded, and concrete when I needed to be. Most importantly, my wife Ivy for being there for me through the highs and the lows, when I prematurely thought my thesis was complete, and when I thought my results were completely in shambles. I couldn’t have done it with- out her. v TABLE OF CONTENTS Biographical Sketch . iii Dedication . iv Acknowledgements . .v Table of Contents . vi List of Symbols . vii 1 Introduction 1 1.1 Motivation from classical analytic number theory . .1 1.2 Discrepancy and the Riemann hypothesis for elliptic curves . .3 2 Discrepancy 7 2.1 Equidistribution . .7 2.2 Definitions and first results . .9 2.3 Statistical heuristics . 12 2.4 The Koksma–Hlawka inequality . 13 2.5 Comparing and combining sequences . 14 2.6 Examples . 19 3 Dirichlet series with Euler product 23 3.1 Definitions and motivation . 23 3.2 Automorphic and motivic L-functions . 27 3.3 Discrepancy and the Riemann hypothesis . 28 4 Irrationality exponents and CM abelian varieties 31 4.1 Definitions and first results . 31 4.2 Irrationality exponents and discrepancy . 34 4.3 Pathological Satake parameters for CM abelian varieties . 37 5 Pathological Galois representations 43 5.1 Notation and supporting results . 43 5.2 Galois representations with specified Satake parameters . 47 5.3 Galois representations with specified Sato–Tate distributions . 51 6 Concluding remarks and future directions 57 6.1 Fake modular forms . 57 6.2 Dense free subgroups of compact semisimple groups . 57 Bibliography 60 vi LIST OF SYMBOLS 1S characteristic function of a set S. l rational prime > 5. −1 f∗m pushforward measure ( f∗m)(S) = m( f (S)). <z real part of z. < > a half-plane fz 2 C : <z > ag. f g there is a constant C > 0 such that f 6 Cg. f (x) xa+e for all e > 0, f (x) xa+e (the constant may depend on e). dx Dirac measure concentrated on x. frp conjugacy class of arithmetic Frobenius at p. G\ space of conjugacy classes of a group G. a a 0 ( b ) shorthand for 0 b . 2 2 ST Sato–Tate measure p sin q dq on [0, p]. x sequence (x1, x2, x3,... ) or (x2, x3, x5,... ). ~x sequence of vectors (~x1,~x2,... ) or (~x2,~x3,~x5,... ). C(X) continuous, C-valued functions on X. f f = o(g) means lim sup g = 0. j · j¥ supremum norm. [~x,~y) half-open box [x1, y1) × · · · × [xd, yd). m[a, b] shorthand for m([a, b]) if m is a measure. D(m, n) discrepancy between m and n. Px,N empirical measure associated to the set fxaga6N. DN(x, m) discrepancy between Px,N and m. Td d-dimensional real torus (R/Z)d. Var( f ) total variation of f . dm dl Radon–Nikodym derivative of m. cdfm(x) cumulative distribution function x 7! m[−¥, x]. x o y interleaved sequence (x1, y1, x2, y2,... ). f = Q(g) there exist constants 0 < C1 < C2 such that C1g 6 f 6 C2g. M x6N : a shorthand for (x1,..., xN, a,..., a) (M copies of a). iq (( + ) ) U (q) tr symk e = sin k 1 q on SU(2)\ = [0, p]. k e−iq sin q L(x, s) Dirichlet series associated to a sequence x in C. L(r(x), s) Dirichlet series associated to a representation. wi(~x) i-th irrationality measure of ~x. h·, ·i standard inner product on Rd. r(m~ ) shorthand for max(1, jm1j) ··· max(1, jmjd). f f = W(g) means lim sup g > 0 (Hardy–Littlewood convention). RF/Q Gm Weil restriction of scalars of Gm. i i H (F, M) Galois cohomology H (GF, M). i XS(M) Tate–Shafarevich group of M. M∗ Cartier dual of M. 1 1 Hnr(Qp, M) unramified cohomology classes in H (Qp, M). vii CHAPTER 1 INTRODUCTION 1.1 Motivation from classical analytic number theory Start with an old problem central to number theory: counting prime numbers. ( ) ( ) = R x dt As usual, let p x be the prime counting function and Li x 2 log t be the Eu- lerian logarithmic integral. The prime number theorem tells us that as x ! ¥, p(x) ! Li(x) 1. The standard approach to proving the prime number theorem is by showing that the Riemann z-function has non-vanishing meromorphic contin- uation to < = 1. Theorem 1.1.1. The function z(s) admits a non-vanishing meromorphic continuation < = = p(x) = to 1 with a simple pole at s 1, if and only if limx!¥ Li(x) 1. Since z(s) does have the desired properties, the prime number theorem is true. It is natural to try to bound the difference p(x) − Li(x). Numerical exper- 1 +e iments dating back to Gauss suggest that jp(x) − Li(x)j x 2 . By this we mean that the estimate holds for any e > 0, though the implied constant may depend on e. In fact, we have the following result. Theorem 1.1.2 ([Edw74, Th., p. 90]). The Riemann hypothesis is true if and only if 1 +e jp(x) − Li(x)j x 2 . Neither side of this equivalence is known for certain to be true! There is an analogue of the above discussion for Artin L-functions. Let K/Q be a nontrivial finite Galois extension with group G = Gal(K/Q). For any 1 rational prime p at which K is unramified, let frp be the conjugacy class of the Frobenius at p in G.
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