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The Cohen-Lenstra Heuristics and Soundararajan's thesis

Sheng, Yan (2016)
Honors Thesis (37 pages)
Committee Chair / Thesis Adviser: Ono, Ken
Committee Members: Duncan, John F ; Moore, Judy Raggi ; Zureick-Brown, David M
Research Fields: Mathematics
Keywords: class number; Cohen-Lenstra heuristics; torsion elements; Diophantine conditions; class group
Program: College Honors Program, Mathematics
Permanent url: http://pid.emory.edu/ark:/25593/rgn2z

Abstract

In this paper, we give an exposition of Kannan Soundararajan's Princeton Ph.D. thesis. His main theorem gives lower bounds on the number of torsion elements of the ideal class group CL(K) for imaginary quadratic fields K = Q( √ −d). The proof relies on counting the number of square free d satisfying certain Diophantine conditions. These conditions are shown to be sufficient for the existence of elements of order g. Proofs of certain classical results from algebraic number theory, such as the finiteness of CL(K), are also included.

Table of Contents

CONTENTS

1. Introduction and statement of results .......................................................1

2. Preliminaries .......................................................................................5

2.1. Elementary Diophantine conditions .......................................................14

3. Proof of Theorem 1.3 ...........................................................................16

3.1. Outline of ideas ................................................................................18

3.2. Counting arguments ..........................................................................20

References ............................................................................................29

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