Split Jacobians and Lower Bounds on Heights
This thesis deals with properties of Jacobians of genus two curves that cover elliptic curves.
- M. Djukanovic
- 01 november 2017
- Thesis in Leiden Repository
This thesis deals with properties of Jacobians of genus two curves that cover elliptic curves. If E is an elliptic curve and C is a curve of genus two that covers it n-to-1 so that the covering does not factor through an isogeny, then C also covers another elliptic curve n-to-1 in such a way and the Jacobian of C is isogenous to the product of the two elliptic curves. The Jacobian is said to be (n,n)-split and the elliptic curves are said to be glued along their n-torsion. The first chapter deals with the geometric aspects of this setup. We describe two approaches to constructing (n,n)-split Jacobians and we investigate which curves can appear in the setup. The second chapter deals with the arithmetic aspects, focusing on height functions and the Lang-Silverman conjecture in particular. We show that this conjecture holds for families of (n,n)-split Jacobians if and only if it holds for the corresponding families of elliptic curves that can be glued along their n-torsion.