This course is an introduction to Riemann surfaces. Starting with examples like the Riemann sphere and the complex tori, we set up the basics of theory (ramification, divisors, differential forms, genus...) in order to tackle the important theorems (Riemann-Hurwitz, Riemann-Roch...). We end up with an opening to important concepts for number theory: modular curves and monodromy theory.

• Definition of Riemann surfaces.
• The Riemann sphere, elliptic functions, complex tori.
• Holomorphic maps between Riemann surfaces.
• Ramification theory, Riemann-Hurwitz formula.
• Meromorphic functions, Riemann-Roch theorem.
• Differential forms, genus of a Riemann surface.
• Quotients of Riemann surfaces. Statement of the uniformization theorem.
• Modular curves and monodromy representations.