David Urbanik, Department of Pure Mathematics, University of Waterloo
"Explicit Equations for Hyperelliptic Jacobians and Their Group Law"
Abelian varieties are groups whose set and whose group law are described by polynomial and rational functions. The simplest example is an elliptic curve, whose set is described by a single degree three projective equation and whose group law is obtained using the well-known chord-and-tangent rule. Higher dimensional generalizations have been extensively studied using abstract methods, but for dimension greater than two no explicit equations for an abelian variety group law are known.
In this talk we give the derivation of explicit equations for the abelian variety group law on hyperelliptic Jacobians of every dimension g. This extends a construction of Mumford which gives explicit equations for the variety itself. Thus we give explicit polynomial and rational functions defining a family of abelian varieties and their group law in every positive dimension.
P.S. The talk is expected to last around one hour.
MC 5417