On Elliptic Curves, the ABC Conjecture, and Polynomial Threshold Functions A dissertation ...

211 Pages·2011·0.8 MB·English

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
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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
3 The Curve Associated to 2A . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2
3.1 The Group A and its Representations . . . . . . . . . . . . . . . . . 82
2
3.2 A Canonical Definition of 2A . . . . . . . . . . . . . . . . . . . . . . 82
2
3.3 The Deligne-Lusztig Curve . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4 Divisors of Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4 The Curve Associated to 2B . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2
4.1 The Group B and its Representations . . . . . . . . . . . . . . . . . 85
2
4.2 A Canonical Definition of 2B . . . . . . . . . . . . . . . . . . . . . . 85
2
4.3 The Deligne-Lusztig Curve . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4 Divisors of Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5 The Curve Associated to 2G . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2
5.1 Description of 2G and its Representations . . . . . . . . . . . . . . . 92
2
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
Ifindmyselfwithanumberofresultsonseparatetopics, havinglittlerelationtooneanother.
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
pairwiserelativelyprime,highlydivisibleintegers. TheABCConjecturestatesthatif“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 be as canonical as possible, with a category-
theoretic construction used at the core of my construction of the twisted Chevalley groups.
In addition to finding models for the curves associated to 2A , 2B and 2G , I also produce
2 2 2
explicit isomorphisms C/Gσ → P1.
Part II consists of Chapter E, which is based upon [30]. In it I deal with the construction
1
CONTENTS 2
of designs of small size in a number of contexts. I find a construction for producing small
designs by making use of a fixed-point theorem from algebraic topology. The bound on size
turns out to be particularly nice in the case of designs on a homogeneous space. I also use a
variant of this technique to find designs of nearly optimal size for the Beta distribution and
on the sphere.
Part III deals with three results on polynomial threshold functions. It begins with an
introduction to the topic providing common motivational background for the topic and
discussing a couple of common themes in this work. It is suggested that the reader look at
this introductory material before reading any of the associated chapters.
Chapter F is based on [27], and proves that limited independence is enough to fool poly-
nomial threshold functions of Gaussians. The proof is based on a technique that I have
been developing along with Jelani Nelson and Ilias Diakonikolas called the FT-Mollification
method, which deals with problems involving k-independence by combining techniques from
approximation theory with anticoncentration results. The main new innovation in this work
is a structure theorem for polynomials that allows us to write them in terms of other poly-
nomials each with small higher moments.
Chapter G is based on [32], and contains proofs of asymptotically optimal bounds on
the Gaussian surface are and noise sensitivity of polynomial threshold functions. The bound
on noise sensitivity is very short and consists largely of taking advantage of a particular
symmetry. The bound on surface area is then derived from the noise sensitivity bound using
a few simple ideas which unfortunately need to be supported by some non-trivial technical
results.
ChapterHis based on [31], and introduces a new pseudorandom generator forpolynomial
threshold functions. Many of the ideas behind this result had their origins in the proofs from
the previous two Chapters.