Physical Sciences and Mathematics Commons^™

Relations Between Theta Functions Of Genus One And Two From Geometry, Thomas Hill

Undergraduate Honors Capstone Projects

Genus-two curves with special symmetries are related to pairs of genus-one curves by two and three-sheeted ramified coverings. This classical work dates back to early 20th century and is known as Jacobi and Hermite reduction. Jacobians of genus-two curves can be used to construct complex two-dimensional complex projective manifolds known as Kummer surfaces. On the other hand, the defining coordinates and parameters of both elliptic curves and Kummer surfaces can be related to Riemann Theta functions and Siegel Theta functions, respectively. This result goes back to the seminal work of Mumford in the 1980s. We use the geometric relation between …