Elliptic and modular functions
Prerequisiti
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.
Programma
- 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.
Obiettivi formativi
The topic of elliptic and modular functions has a major historical importance, is linked with many other central mathematical topics, and is also nowadays of the maximum attraction for researchers in many areas. Therefore it is fundamental that every mathematician has at least a basic background knowledge, and to furnish this knowedge is the main object of the course.
Riferimenti bibliografici
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