Regularized Integrals on Riemann Surfaces and Modular Forms

Modular forms and K3 surfaces

For every known Hecke eigenform of weight 3 with rational eigenvalues we exhibit a K3 surface over Q associated to the form. This answers a question asked independently by Mazur and van Straten. The proof builds on a classification of CM forms by the second author.

Iterated Integrals of Modular Forms and Noncommutative Modular Symbols

The main goal of this paper is to study properties of the iterated integrals of modular forms in the upper halfplane, eventually multiplied by zs−1, along geodesics connecting two cusps. This setting generalizes simultaneously the theory of modular symbols and that of multiple zeta values. §0. Introduction and summary This paper was inspired by two sources: theory of multiple zeta values on the...

On Cycle Integrals of Weakly Holomorphic Modular Forms

In this paper, we investigate cycle integrals of weakly holomorphic modular forms. We show that these integrals coincide with the cycle integrals of classical cusp forms. We use these results to define a Shintani lift from integral weight weakly holomorphic modular forms to half-integral weight holomorphic modular forms.

Moduli spaces of isoperiodic forms on Riemann surfaces

This paper describes the global geometry of the moduli space of holomorphic 1-forms (X,ω) with fixed periods on varying Riemann surfaces of genus g. It establishes completeness for general g and then explores several features of the case g = 2, which yields the first detailed, global picture of the transverse dynamics of the Teichmüller geodesic flow.

On the Defining Equations of Abelian Surfaces and Modular Forms