Doubly periodic functions. Lattices, complex tori. The field of elliptic functions.
Riemann surfaces: definitions, maps, the genus, Riemann–Hurwitz formula.
Differential forms on a Riemann surface. The Abel–Jacobi map.
Local study of holomorphic functions. Points of Riemann surfaces as valuations on the field of meromorphic functions. Classifications of the absolute values on the field of rational numbers. The field of p-adic numbers. The product formula.
Algebraic curves over a field. Algebraic curves over C and relation to Riemann surfaces.
Rational points. Arithmetic of curves according to the genus. The case of conics (genus 0). Hasse principle. Finite generation of the rational points on elliptic curves (genus 1): the case of Fermat quartic.
Modular surfaces and modular forms. Analytic construction of rational points on elliptic curves.