Universiteit Leiden

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Dissertation

Split Jacobians and Lower Bounds on Heights

This thesis deals with properties of Jacobians of genus two curves that cover elliptic curves.

Author
M. Djukanovic
Date
01 November 2017
Links
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.