Alexander polynomials of curves and Mordell-Weil ranks of Abelian threefolds

Algebra Seminar
Friday, January 10, 2014 - 3:05pm
1 hour (actually 50 minutes)
Skiles 006
Humboldt University Berlin
Let $C=\{f(z_0,z_1,z_2)=0\}$ be a complex plane curve with ADE singularities. Let $m$ be a divisor of the degree of $f$ and let $H$ be the hyperelliptic curve $y^2=x^m+f(s,t,1)$ defined over $\mathbb{C}(s,t)$. In this talk we explain how one can determine the Mordell-Weil rank of the Jacobian of $H$ effectively. For this we use some results on the Alexander polynomial of $C$. This extends a result by Cogolludo-Augustin and Libgober for the case where $C$ is a curve with ordinary cusps.  In the second part we discuss how one can do a similar approach over fields like $\mathbb{Q}(s,t)$ and $\mathbb{F}(s,t)$.