Shintani lifts for weakly holomorphic modular forms

点击翻译按钮获取中文摘要

Modular forms have played a pivotal role in the proof of many groundbreaking theorems, including Wiles's proof of Fermat's Last Theorem and Tunnell's conditional solution to the congruent number problem. A key ingredient in the proof of Tunnell's theorem is the Shintani lift and its adjoint Shimura lift, which map between integral and half-integral weight modular forms.A natural generalization of modular forms is weakly holomorphic modular forms, which exhibit certain weaker growth conditions than classical modular forms. Weakly holomorphic modular forms play an important role in various areas of mathematics and physics, including, for example, Monstrous Moonshine (relating dimensions of irreducible representations of the monster group to Fourier coefficients of the modular j-function). The aim of this project is to extend the Shintani lift to map between spaces of integral and half-integral weight weakly holomorphic modular forms and to understand its applications and interactions with the Hecke operators.