Examination procedure
<p> oral exam.</p>
Prerequisites
The prerequisites for this course are modest: masic complex analysis in one variable, basic algebra.Some notions of Algebraic Gometry will be given during the course.Therefore the course is accessible to students from the III year onwards.
Syllabus
- Generalities, Riemann surfaces.Meromorphic doubly periodic functions; theorems of Liouville, Weierstrass functions, addition theorems.
- Elliptic functions and complex elliptic curves, isomorphism classes, endomorphisms and automorphisms, points of finite order, isogenies. Complex Multiplication.
- The modular group, fundamental domain, generation.
- Modular functions and forms, Eisenstein series, the modular form Delta and elliptic curves, spaces of modular forms of given weight.
- The modular invariant j, singular invariants and special values, modular equations.
- Fourier expansions, order of growth of coefficitns of modular forms.
Bibliographical references
For general basic theory of Riemann Surfaces: the first chapters of Milnor's "dynamics in one complex variable" are perfect.For elliptic functions: Lang's "Elliptic Functions" is perfect.Other suitable books are Silverman's "The arithmetic of elliptic curves", especially the appendix on algebraic geometryFurther bibliography will be suggested during the course