##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Djordjo Milovic

#### Univ. Leiden and Univ. Paris-Sud 11

## Density results on the $2$-part of class groups

##### Abstract:

We will discuss some new density results about the $2$-primary part of class groups of quadratic number fields and how they fit into the framework of the Cohen-Lenstra heuristics. Let $\mathrm{Cl}(D)$ denote the class group of the quadratic number field of discriminant $D$. The first result is that the density of the set of prime numbers $p\equiv -1\bmod 4$ for which $\mathrm{Cl}(-8p)$ has an element of order $16$ is equal to $1/16$. This is the first density result about the $16$-rank of class groups in a family of number fields. The second result is that in the set of fundamental discriminants of the form $-4pq$ (resp. $8pq$), where $p\equiv q \equiv 1\bmod 4$ are prime numbers and for which $\mathrm{Cl}(-4pq)$ (resp. $\mathrm{Cl}(8pq)$) has $4$-rank equal to $2$, the subset of those discriminants for which $\mathrm{Cl}(-4pq)$ (resp. $\mathrm{Cl}(8pq)$) has an element of order $8$ has lower density at least $1/4$ (resp. $1/8$). We will briefly explain the ideas behind the proofs of these results and emphasize the role played by general bilinear sum estimates. \newline\newline Note: The speaker will give a prep-talk for graduate students in AP&M 7421 at 1:15pm. All graduate students interested in number theory are strongly encouraged to attend.

Host: Cristian Popescu

### January 21, 2016

### 1:00 PM

### AP&M 7321

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