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

Sheng, Yan (2016)

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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

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