The Cohen-Lenstra Heuristics and Soundararajan's thesis Open Access

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

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

About this Honors Thesis

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• Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.

School	Emory College
Department	Mathematics
Degree	BS
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	class number Cohen-Lenstra heuristics Diophantine conditions class group torsion elements
Committee Chair / Thesis Advisor	Ono, Ken, Emory University
Committee Members	Moore, Judy Raggi, Emory University Zureick-Brown, David M, Emory University Duncan, John F, Emory University

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