Representation of SL2Z

This lecture explores the representation of the group SL2Z, which naturally acts on a two-dimensional vector space and the integer lattice Z2. The instructor defines a function F on the dual lattice quotiented by the lattice, showing its transformation properties and its relation to modular forms. The lecture delves into the computation of the function, including the Gauss sum, and its application in understanding the transformation law of theta functions. The instructor demonstrates how the function F behaves under translations and other generators of the full modular group, highlighting its significance in understanding the transformation properties of theta functions.