Shintani lifts for weakly holomorphic modular forms

模形式在许多开创性定理的证明中发挥了关键作用，包括怀尔斯对费马大定理的证明和通内尔对全等数问题的条件解。 Tunnell定理证明的一个关键要素是Shintani提升和它的伴随Shimura提升，它们映射在整数和半整数权模形式之间。模形式的自然推广是弱全纯模形式，它表现出比经典模形式更弱的增长条件。 弱全纯模形式在数学和物理学的各个领域都扮演着重要的角色，例如Monte-Moonshine（将怪物群的不可约表示的维数与模j函数的傅里叶系数联系起来）。 这个项目的目的是将Shintani提升扩展到积分和半积分权弱全纯模形式空间之间的映射，并理解其应用和与Hecke算子的相互作用。

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.