Modular Lambda Function: Properties and Applications

This lecture introduces the modular lambda function and its properties, focusing on its role as a Hauptmodul for Gamma(2). The instructor explains how the function is invariant under the action of Gamma(2), has no poles in the upper half plane, and provides a holomorphic bijection between the quotient space and the complex projective line without three points. The lecture also covers the computation of zeros of the function inside the fundamental domain of Gamma(2) and its significance as a Hauptmodul for the group. Through detailed proofs and computations, the instructor demonstrates the transformation properties of the lambda function and its relation to modular forms of weight zero, highlighting its importance in the theory of modular curves and Riemann surfaces.