Dissertation Defense Seminar

This dissertation applies Vinberg theory to the problem of constructing 2-descent maps on the Jacobians of hyperelliptic curves. In the first part, we construct, given a hyperelliptic curve $C$ with certain marked points $P$ and tangent vectors $t$, a smooth, complete surface $S$. Using the Picard group of $S$, we obtain a 2-graded simple, split adjoint Lie group $(H,\theta)$ of Dynkin type $A_n,$ where $n$ depends on the genus of $C$ and the nature of the marked points. We show that there exists an injective 2-descent map from the Jacobian of $C$ into the orbit spaces of the local and global Vinberg representations associated to $(H,\theta).$ The approach is based on previous work of Thorne which addresses the cases where $H$ is of Dynkin type $E_6$ or $E_7.$


In the second part, we construct, given a polynomial $f(x)$ of odd degree $n$ and satisfying certain properties, an explicit map $\psi$ from $J(C)$ into the orbit space of $G = \SO(V) \times \SO(V)$ acting on $\End(V),$ where $C$ is the smooth, complete curve satisfying the equation $y^2=f(x),$ and $V$ is an orthogonal space of dimension $n$ with maximal index of isotropy and discriminant $1.$ The pair $(G,V)$ is exactly the degree 2 Vinberg representation of Dynkin type $D_n.$ The map $\psi$ extends a previous construction of Thorne which gives an explicit descent map for the degree 2 Vinberg representation of Dynkin type $A_n.$

In both parts, we work over an arbitrary field $K$ of characteristic 0.

Jason's advisor is Wei Ho. Speaker(s): Jason Liang (UM)