Hi everybody! Me (Anton) and Stanley are launching the seminar on modular forms this semester! Should be tons of psychedelic fun. Our basic goal for this seminar is to figure out whats so special about modular forms and why are they incredibly useful in various areas of number theory. But of course, being incredibly ambitious individuals, we also dream to understand the statement of the modularity theorem, modular curves and moduli spaces, and (fingers crossed) automorphic forms and Shimura varieties, which (hopefully) we will study next semester.
CONTENTS OF SEMINAR 1: We will start with basic definitions and examples of modular forms. In particular, we will look at the Eisenstein’s series and, just to get a feeling of how the modular forms are related to the number theory, we’ll see the modular forms associated to the number of integer solutions (t,x,y,z) to n=t^2+x^2+t^2+z^2 for some fixed n.