COEFFICIENTS of HALF INTEGRAL WEIGHT MODULAR FORMS Mathcs Emory

Instructions and help about COEFFICIENTS OF HALF INTEGRAL WEIGHT MODULAR FORMS Mathcs Emory

this lecture is part of an online series of lectures about modular forms and it's the introductory lecture where i will explain roughly what a modular form is and try and explain why they're interesting so first of all we have the definition of a modular form and the definition doesn't look very promising it says that f of a tau plus b over c tau plus d is equal to c tau plus d to the k for some k times f of tau here a b c d is a two by two matrix with integer coefficients with determinant one so sl2 means determinant one and this looks like you know you can imagine you can could write down dozens of similar definitions of a function equation and it's not at all clear why these should be of any particular interest a special case when k equals zero is called a modular function so modular functions um which you might think are the simplest sorts of modular forms are just invariant under this group action um so they look like that here um tau is a complex number whose imaginary part is gr