# Elliptic Functions and Modular Forms

Lecturer: Silviu Radu

Abstract: In the course we prove the basic theorems about compact Riemann surfaces. We state the Riemann-Roch theorem. For the rest of the course we concentrate on compact Riemann surfaces that arise as orbits of a group acting on a subset of the complex plane. In particular we look at the action of the modular group on the upper half complex plane. The meromorphic functions on this compact Riemann surface are in one to one correspondence with the modular functions. Similarly the orbits of the action of a lattice L on the complex plane gives rise to a compact Riemann surface T (also known as the Torus). The meromorphic functions on this compact Riemann surface are in one to one correspondence with the elliptic functions invariant under L. In this way we make use of general theory about compact Riemann surfaces to deduce results about elliptic functions and modular functions (from which we deduce results about modular forms).

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