Riemann-Roch and Elliptic Curves

The Riemann-Roch theorem (for compact Riemann surfaces) is a deep connection between complex analysis and topology. It states roughly that the behavior of functions, with prescribed analytic properties, on a surface is constrained by the genus of that surface. We prove this theorem and give an application – that nonsingular projective algebraic curves of genus one (e.g. elliptic curves) are described by cubic equations.