Seminar Elliptic Functions and Modular Forms

We start with the basic theory of elliptic functions, which are doubly periodic meromorphic functions on the complex plane. As a fundamental example we will construct the Weierstrass p-function, which serves as a building block for all elliptic functions, and which parametrizes complex elliptic curves. We will also discuss the Weierstrass sigma and zeta functions and the Jacobi theta function, which can be used to construct elliptic functions with prescribed zeros and poles.

From elliptic functions one is naturally led to the study of modular forms. These are holomorphic functions on the complex upper half-plane that have infinitely many symmetries under Moebius transformations. Using the residue theorem one can show the remarkable fact that the vector spaces of modular forms of fixed weight are finite-dimensional. As examples of modular forms we will study Eisenstein series, the Delta-function, and the j-invariant. By computing the Fourier expansions of modular forms and using the finite-dimensionality of the spaces of modular forms, one obtains interesting applications in number theory. For example, we will prove the fact that every natural number can be written as a sum of four squares. At the end of the seminar we will define Hecke operators and use them to prove that the Fourier coefficients of the Delta-function are multiplicative. Finally, we study the L-functions associated with modular forms and prove their analytic continuation and functional equation.