Vinberg Representations and 2-Descent on Jacobians of Curves

Abstract

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 and tangent vectors, a smooth, complete surface S. Using the Picard group of S, we obtain a 2-graded simple adjoint Lie group H of Dynkin type An, where n depends on the genus of C and the nature of the marked points. Assuming that H is split, 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. The approach is based on previous work of Thorne which addresses the cases where H is of Dynkin type E6 or E7.

In the second part, we construct, given a polynomial f(x) of odd degree n and satisfying certain properties, an explicit map from J(C) into the orbit space of the Dynkin representation (G,V) of type Dn, where C is the hyperelliptic curve with Weierstrass points at the roots of f(x). Our map extends a previous construction of Thorne which gives an explicit descent map for the degree 2 Vinberg representation of Dynkin type An.

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