Arithmetic intersection theory and automorphic forms

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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.