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On Elliptic Curves, the ABC Conjecture, and Polynomial Threshold Functions

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On Elliptic Curves, the ABC Conjecture, and Polynomial Threshold Functions On Elliptic Curves, the ABC Conjecture, and Polynomial Threshold Functions A dissertation presented by Daniel Mertz Kane to The Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics Harvard University Cambridge, Massachusetts March, 2011 © 2011{Daniel Mertz Kane All rights reserved. iii Dissertation Advisor: Professor Barry Mazur Daniel Mertz Kane On Elliptic Curves, the ABC Conjecture, and Polynomial Threshold Functions Abstract We present a number of papers on topics in mathematics and theoretical computer sci- ence. Topics include: a problem relating to the ABC Conjecture, the ranks of 2-Selmer groups of twists of an elliptic curve, the Goldbach problem for primes in specified Cheb- otarev classes, explicit models for Deligne-Lusztig curves, constructions for small designs, noise sensitivity bounds for polynomials threshold functions, and pseudo-random generators for polynomial threshold functions. Contents I Number Theory 3 A On A Problem Relating to the ABC Conjecture 4 1 Introduction .................................... 4 2 Lattice Methods .................................. 5 2.1 Lattice Lower Bounds .......................... 5 2.2 Diadic Intervals .............................. 7 2.3 Lattice Upper Bounds .......................... 7 3 Points on Conics ................................. 10 4 Conclusion ..................................... 12 B The Goldbach Problem for Primes in Chebotarev Classes 13 1 Introduction and Statement of Results ..................... 13 2 Overview ...................................... 16 2.1 The Proof of Theorem 1 ......................... 16 2.2 Outline of Our Proof ........................... 17 3 Preliminaries ................................... 18 3.1 G and F .................................. 18 3.2 Local Approximations .......................... 20 3.3 Smooth Numbers ............................. 23 4 Approximation of F ............................... 23 4.1 α Smooth ................................. 24 4.1.1 Results on L-functions ..................... 25 4.1.2 Approximation of F ...................... 26 4.1.3 Approximation of F ] ...................... 26 4.1.4 Proof of Proposition 7 ..................... 28 4.2 α Rough .................................. 28 4.2.1 Bounds on F ] .......................... 28 4.2.2 Lemmas on Rational Approximation . 30 4.2.3 Lemmas on Exponential Sums . 34 4.2.4 Bounds on F .......................... 36 iv CONTENTS v 4.3 Putting it Together ............................ 39 5 Approximation of G ............................... 40 6 Proof of Theorem 2 ................................ 41 6.1 Generating Functions ........................... 41 6.2 Dealing with H] .............................. 43 6.3 Putting it Together ............................ 45 7 Application .................................... 46 C Ranks of 2-Selmer Groups of Twists Elliptic Curves 48 1 Introduction .................................... 48 2 Preliminaries ................................... 51 2.1 Asymptotic Notation ........................... 51 2.2 Number of Prime Divisors ........................ 51 3 Character Bounds ................................. 52 4 Average Sizes of Selmer Groups ......................... 66 5 From Sizes to Ranks ............................... 72 D Projective Embeddings of the Deligne-Lusztig Curves 76 1 Introduction .................................... 76 2 Preliminaries ................................... 77 2.1 Basic Theory of the Deligne-Lusztig Curve . 77 2.2 Representation Theory .......................... 79 2.3 Basic Techniques ............................. 80 2.4 Notes ................................... 81 2 3 The Curve Associated to A2 .......................... 82 3.1 The Group A2 and its Representations . 82 2 3.2 A Canonical Definition of A2 ...................... 82 3.3 The Deligne-Lusztig Curve ........................ 82 3.4 Divisors of Note .............................. 83 2 4 The Curve Associated to B2 .......................... 85 4.1 The Group B2 and its Representations . 85 2 4.2 A Canonical Definition of B2 ...................... 85 4.3 The Deligne-Lusztig Curve ........................ 88 4.4 Divisors of Note .............................. 88 2 5 The Curve Associated to G2 .......................... 92 2 5.1 Description of G2 and its Representations . 92 5.2 Definition of σ ............................... 93 5.3 The Deligne-Lusztig Curve ........................ 97 5.4 Divisors of Note .............................. 98 CONTENTS vi II Designs 102 E Designs for Path Connected Spaces 103 1 Introduction .................................... 103 2 Basic Concepts .................................. 104 3 The Bound for Path Connected Spaces . 107 4 Tightness of the Bound .............................. 111 5 The Bound for Homogeneous Spaces . 113 5.1 Examples ................................. 115 5.2 Conjectures ................................ 115 6 Designs on the Interval .............................. 115 7 Spherical Designs ................................. 123 III Polynomial Threshold Functions 137 1 Introduction .................................... 138 F k-Independence Fools Polynomial Threshold Functions 141 1 Introduction .................................... 141 2 Overview ...................................... 142 3 Moment Bounds .................................. 144 4 Structure ..................................... 146 5 FT-Mollification .................................. 149 6 Taylor Error .................................... 152 7 Approximation Error ............................... 155 8 General Polynomials ............................... 160 9 Conclusion ..................................... 162 G Noise Sensitivity of Polynomial Threshold Functions 164 1 Introduction .................................... 164 1.1 Background ................................ 164 1.2 Measures of Sensitivity . 164 1.3 Previous Work .............................. 166 1.4 Statement of Results . 167 1.5 Application to Agnostic Learning . 167 1.5.1 Overview of Agnostic Learning . 167 1.5.2 Connection to Gaussian Surface Area . 168 1.6 Outline .................................. 168 2 Proof of the Noise Sensitivity Bound . 168 3 Proof of the Gaussian Surface Area Bounds . 170 CONTENTS vii 4 Conclusion ..................................... 173 H A PRG for Polynomial Threshold Functions 180 1 Introduction .................................... 180 2 Definitions and Basic Properties . 182 3 Noisy Derivatives ................................. 184 4 Averaging Operators ............................... 187 5 Proof of Theorem 1 ................................ 191 6 Finite Entropy Version .............................. 195 Acknowledgements I would like to thank those who have been influential in my mathematics education, namely Jonathan Kane, Ken Ono, the organizers of MOP, David Jerison, Victor Guillemin, Erik Demaine, Noam Elkies, Benedict Gross, Barry Mazur, and Henry Cohn. I would also like to thank Jelani Nelson and Ilias Diakonikolas for getting me involved in much of my recent computer science work. The material in Chapter E was developed while working as a summer intern at Microsoft Research New England. I am also grateful for the NSF and NDSEG graduate fellowships which have helped support me financially during graduate school. viii Introduction My work in the past few years has been largely centered around problem solving. As a result I find myself with a number of results on separate topics, having little relation to one another. As such, this thesis consists of a number of largely independent results. Each Chapter in this thesis (with the exception of those in Part III, which share motivational background) is meant to stand on its own. Part I contains several results in the field of number theory, mostly of the analytic variety. Chapter A is based on [28] and studies a problem related to the ABC Conjecture. In particular I look at the number of solutions to the equation A + B + C = 0 for A; B; C pairwise relatively prime, highly divisible integers. The ABC Conjecture states that if \highly divisible" is defined strictly enough that there should be only finitely many such triples. In this Chapter, I obtain asymptotic results for the number of such solutions for a broad range of weaker meanings of \highly divisible". My major techniques are the use of lattice methods and bound do to Heath-Brown on the density of integer points on a conic. Chapter B is based on [25], and deals with a variant of the Goldbach problem. In particular, I prove an asymptotic for the number of solutions to a linear equation in at least three prime numbers given that these primes are required to lie in specified Chebotarev classes. I make use of a number of analytic techniques including Hecke L-functions, the circle method and Vinogradov's method. Chapter C is based on [29], and deals with the ranks of 2-Selmer groups of twists of an elliptic curve. In particular, I show that for a wide range of elliptic curves, Γ, that the ranks of the 2-Selmer groups of their twists satisfy a particular distribution. This extends a result of Swinnerton-Dyer ([58]), which gives the same asymptotics but with an unusual notion of density. I extend Swinnerton-Dyer's result to use the standard notion of density using a number of analytic techniques including L-functions and the large sieve. Chapter D is based on [26], and produces explicit models for the Deligne-Lusztig varieties of dimension 1. The constructions attempt to
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