Arithmetic of Elliptic Curves and Abelian Varieties over Function Fields

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The study of elliptic surfaces over finite fields, or equivalently, of elliptic curves defined over function fields of transcendence degree one over finite fields is a venerable and highly developed area of arithmetic geometry. It is important testing area for central conjectures in the subject, such as the Tate, Beilinson and Parshin conjectures, but is also a rich source for discovering new phenomena. One of the major research directions of the area to arise in the last two decades is the study of these elliptic curves with the help of modular parameterizations by Drinfeld modular curves. Such parameterizations were successfully used to show special cases of the conjectures above, or prove other, more unexpected results. These developments required, and hence lead to a much finer understanding of these modular parameterizations, starting with the foundational work of Gekeler and Reversat. Not surprisingly the study of these parameterizations is now a very important subject of its own, with its own central problems. In this project we aim to resolve two central outstanding conjectures in the theory of modular parameterizations of elliptic curves by Drinfeld modular curves, namely Mazur's conjecture on the modular height of strong Weil curves, and the Gekeler-Reversat conjecture on the rigid analytic theta-functions of these modular parameterizations. We also expect that the methods which we will develop in order to resolve these decades-old conjectures will lead to the resolution of the uniform boundedness conjecture of Poonen-Schweizer, too.