##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Maike Massierer

#### University of New South Wales

## Counting points on some geometrically hyperelliptic curves of genus 3 in average polynomial time

##### Abstract:

Let $C/\mathbb{Q}$ be a curve of genus 3, given as a double cover of a conic with no $\mathbb{Q}$-rational points. Such a curve is hyperelliptic over the algebraic closure of $\mathbb{Q}$ but does not have a hyperelliptic model of the usual form over $\mathbb{Q}$. We discuss an algorithm that computes the local zeta functions of $C$ simultaneously at all primes of good reduction up to a given bound $N$ in time $(\log N)^{4+o(1)}$ per prime on average. It works with the base change of $C$ to a quadratic field $K$, which has a hyperelliptic model over $K$, and it uses a generalization of the ``accumulating remainder tree'' method to matrices over $K$. We briefly report on our implementation and its performance in comparison to previous implementations for the ordinary hyperelliptic case. Joint work with David Harvey and Andrew V. Sutherland. In the pre-talk, we will introduce some of the objects that the talk is about, such as curves and their models, the zeta function and how it relates to point counting, and the particular type of genus 3 curves that we are interested in.

Host: Kiran Kedlaya

### February 18, 2016

### 1:00 PM

### AP&M 7321

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