Elliptic and modular functions
Prerequisiti
Complex analysis is a prerequisite, and some elementary number theory.
Programma
- 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.
- Zeta functions, sigma functions and expansions. Theta functions.
- Notions on congruence subgroups, the congruence subgroup problem.
- Modular forms with level and arithmetic applications.
Obiettivi formativi
The course contains basic material from several sources: complex analysis, algebraic geomtery and algebra. It represents an introduction to the theory of elliptic and modular functions, which will contribute to the general culture of any student and allow those particularly interested in the topics to access more sophisticated treatments in the desired directions.
Riferimenti bibliografici
- S. Lang, Elliptic functions, addison Weslay
J-P. Serre, cours d'arithme'tique (last chapter), PUF.
Further references will be suggested during the course,