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

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

$\mathbb{F}(s,t)$ .

Georgia Institute of Technology College of Sciences