Syllabus
Riemann surfaces
Upper half plane and SL(2, R)
Elliptic functions
Modular forms
Eisenstein's series, Ramanujan's tau function
Hecke operators
Zeta function and Dirichlet L-functions
Analytic continuation and functional equation
Theta functions
L-functions of modular forms and elliptic curves
FLT and modularity theorem
Annotation
Modular forms and L-functions are central objects in modern number theory, which played an important role in the proof of Fermat's Last Theorem. They are certain complex functions encoding information of number-theoretic interest, e.g., about the distribution of prime numbers, or numbers of solutions of diophantine equations. Combining analytic and algebraic methods, the course will cover their basic properties and some applications. Specific choice of topics will depend on the interests of participants.
The course may not be taught every academic year.