Arithmetic of Elliptic Curves and Abelian Varieties over Function Fields

有限域上的椭圆曲面的研究，或等价地，有限域上超越次为1的函数场上定义的椭圆曲线的研究，是算术几何中一个古老而高度发达的领域。它是该学科中心猜想（如Tate猜想、Beilinson猜想和Parshin猜想）的重要测试区域，也是发现新现象的丰富来源。利用德林菲尔德模曲线的模参数化来研究这些椭圆曲线是近二十年来该领域兴起的主要研究方向之一。这样的参数化被成功地用于显示上述猜想的特殊情况，或证明其他更意想不到的结果。从Gekeler和Reversat的基础工作开始，这些发展需要并因此导致对这些模块化参数化的更精细的理解。毫不奇怪，这些参数化的研究现在是一个非常重要的主题，有它自己的中心问题。在本项目中，我们旨在解决Drinfeld模曲线椭圆曲线模参数化理论中的两个中心突出猜想，即关于强Weil曲线的模高度的Mazur猜想，以及关于这些模参数化的刚性解析函数的Gekeler-Reversat猜想。我们还期望，为了解决这些几十年来的猜想，我们将发展的方法也将导致Poonen-Schweizer的一致有界猜想的解决。

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.