Some studies on Shimura's half integral weight modular forms

Abstract

Half integral weight modular forms are defined by using a lifting of $GL_2$(Q). Since the Hecke operators are defined on half integral weight modular forms, the Euler products of half integral weight modular forms are obtained. It is shown that half integral weight modular forms are closely related with integral weight modular forms under the Shimra``s map. And as its application, the criterion on classical congruent number problems is obtained.