Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp019c67wq66t
 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