Skip navigation
Please use this identifier to cite or link to this item:
Title: Cohen-Lenstra sums over \(\mathbb{Z}_p[C_p]\)
Authors: Tyler, Matt
Advisors: Bhargava, Manjul
Department: Mathematics
Class Year: 2019
Abstract: The Cohen-Lenstra heuristics predict that the \(p\)-part of the class group of a random imaginary quadratic field is isomorphic to an abelian \(p\)-group \(A\) with probability proportional to \(\frac{1}{|\text{Aut} A|}\). This probability distribution has since arisen in a myriad of different settings, and serves as a natural model for a random abelian group. More generally, the Cohen-Lenstra-Martinet heuristics deal with sums of the form \(\sum_M \frac{1}{|\text{Aut} M|}\) over a collection of \(\mathbb{Z}_p[G]\)-modules \(M\), where \(G\) is a group such that \(p \nmid |G|\). When \(p \mid |G|\), however, things become more complicated, and much less is known. We consider the simplest such case, when \(G\) is the cyclic group \(C_p\) of order \(p\). We study modules over \(\mathbb{Z}_p[C_p]\) and evaluate \(\sum_M \frac{1}{|\text{Aut} M|}\) for various collections of \(\mathbb{Z}_p[C_p]\)-modules \(M\).
Type of Material: Princeton University Senior Theses
Language: en
Appears in Collections:Mathematics, 1934-2020

Files in This Item:
File Description SizeFormat 
TYLER-MATT-THESIS.pdf398.49 kBAdobe PDF    Request a copy

Items in Dataspace are protected by copyright, with all rights reserved, unless otherwise indicated.