Composition Laws

The parametrization of ideal classes of quadratic rings by binary quadratic forms has been an important tool for computing class numbers of quadratic fields. We will discuss how in this classical theorem, the integers can be replaced by the projective line, quadratic rings are then replaced by hyperelliptic curves, and ideal classes are replaced by line bundles on those curves. This gives a very explicit parametrization of line bundles on hyperelliptic curves by certain forms that are “binary quadratic forms over the projective line.