## Abelian Varieties Multi-Site Seminar Series: Nicolas Theriault

- Date: 06/10/2016

*University of Santiago (Chile)*

University of Calgary

Bisections and Squares in Hyperelliptic Curves [video]

For elliptic curves, the Mordell-Weil Theorem allows to relate bisections (pre-images of the multiplication by 2) in the group of points of a curve defined over F_q and the quadratic reciprocity of some elements in the field F_q, which can be used to obtain an algorithm to bisect points in E(F_q). For reduced divisors D=[u(x),v(x)] (in Mumford representation) in the Jacobian of imaginary hyperelliptic curves y^2=f(x) (with f(x) squarefree and of odd degree), we show a relation between the existence of F_q-rational bisections and the quadratic character of u(x) when it is evaluated at the roots of the polynomial f(x) (i.e. at the x-coordinates of the Weierstrass points). This characterization allows us to compute all the bisections of a reduced divisor computing a few square roots (2g square roots if f(x) has 2g+1 roots in F_q) and solving a small system of linear equations.For hyperelliptic curves of genus 2, we obtain an equivalent characterization for curves with a real model (with f(x) squarefree of degree 6) when working with balanced divisors.

A video of this event is available on mathtube.org.

Time: 14/15/17/23 PST/MST/EST/CET

**Bluejeans Connection Info:**

To join from a computer or phone:

Follow this link.

Connecting directly from a room system?

1) Dial: 199.48.152.152 or bjn.vc

2) Enter Meeting ID: 642485886

Just want to dial in on your phone?

1) Direct-dial with my iPhone or

+1.408.740.7256 (US)

+1.888.240.2560 (US Toll Free)

+1.408.317.9253 (Alternate number)

(all numbers)

2) Enter Meeting ID: 642485886

3) Press #