Arithmetic intersection theory and automorphic forms

算术几何，研究数论和几何之间的相互作用， 包括研究多项式方程的整数(整数)解，其中 自古以来就让数学家们着迷。近年来，出现了一些重要的新工具 它的发展使得费马等长期存在的经典问题得以解决 最后一个定理。最强大的现代方法之一涉及到 算术几何和模形式理论，即对算术信息进行编码的分析对象。 这项研究的目标是在识别出这种类型的关系的基础上建立新的关系 模形式的傅里叶系数与高度，一个复杂性的度量， 整数解的集合。这些新的关系揭示了模块化形式在这方面所起的作用 区域。

Arithmetic geometry, the study of the interaction between number theory and geometry, includes the study of whole number (integer) solutions of polynomial equations, which has fascinated mathematicians since antiquity. In recent years, significant new tools have been developed that have allowed the solution of long standing classical problems such as Fermat's last theorem. One of the most powerful modern methods involves the relations between arithmetic geometry and the theory of modular forms, analytic objects which encode arithmetic information. The goal of this research is to establish new relations of this sort based on the identification of the Fourier coefficients of modular forms with the heights, a measure of the complexity, of integer solutions. These new relations shed light on the role played by modular forms in this area.