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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 \(M\). |

URI: | http://arks.princeton.edu/ark:/88435/dsp019c67wq66t |

Type of Material: | Princeton University Senior Theses |

Language: | en |

Appears in Collections: | Mathematics, 1934-2020 |

Files in This Item:

File | Description | Size | Format | |
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TYLER-MATT-THESIS.pdf | 398.49 kB | Adobe PDF | Request a copy |

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