Content

This course is a introduction to the classical theory of modular forms: Geometry of the hyperbolic plane, group of isometries, discrete group actions, Eisenstein series, Poincaré series, Fourier expansions of modular forms, congruence subgroups, Hecke operators, Petersson inner product, estimates for Fourier coefficients, Kloosterman sums and the Petersson formula. We will then move to arithmetic applications of modular forms. Among the many applications of the theory, we will focus on theta functions, representations of integers by quadratic forms, equidistribution of integer points on ellipsoids, and the Riemann zeta function.

Prerequisites

Linear algebra, modern algebra, real analysis, complex analysis, and elementary number theory.

Schedule

Date	Format	Comment
Tuesday 19.09.2017	Lecture	Analytic continuation and functional equation of the zeta function; intro to the theta function and Gauss sums
Thursday 21.09.2017	Lecture	The Jacobi symbol, Gauss sums, quadratic reciprocity, and the automorphy relation for the theta function
Tuesday 26.09.2017	Exercise	Exercises. Solutions.
Thursday 28.09.2017	Lecture	Congruence subgroups, fundamental domains, definition of modular functions
Tuesday 3.10.2017	Lecture	Growth conditions, invariance by a larger group, background on characters of an abelian group
Thursday 5.10.2017	Lecture	Orthogonality relations, nebentype characters, extension to the Hecke-Iwahori subgroup. Examples of modular forms.
Tuesday 10.10.2017	Exercise	Exercises. Solutions.
Thursday 12.10.2017	Lecture	Lagrange's four squares theorem proved via theta functions; Fourier expansions
Tuesday 17.10.2017	Lecture	Fourier expansions, the dimension formula for SL_2(Z)
Thursday 19.10.2017	Lecture	We finish the dimension formula for SL_2(Z), then start hyperbolic geometry
Tuesday 24.10.2017	Exercise	Exercises. Solutions.
Thursday 26.10.2017	Lecture	Hyperbolic metric and measure, the topology on the modular curve, connected locally compact Hausdorff
Tuesday 31.10.2017	Lecture	Stabilizers in SL_2(Z) and elliptic points, compactification and cusps
Thursday 02.11.2017	Lecture	Topology of X(Gamma), local uniformizers, hyperbolic measure on Y(Gamma), Petersson inner product, finite dimensionality
Tuesday 07.11.2017	Exercise	Exercises. Solutions.
Thursday 09.11.2017	Lecture	Finish the proof of finite-dimensionality. Begin Hecke Theory.
Tuesday 14.11.2017	Lecture	Hecke Theory.
Thursday 16.11.2017	Lecture	Continuation of Hecke theory. Eisenstein series and Poincaré series.
Tuesday 21.11.2017	Exercise	Exercises. Solutions.
Thursday 23.11.2017	Lecture	Action of Hecke operators on Poincaré series, Hecke basis
Tuesday 28.11.2017	Lecture	Hecke Theory for Gamma_0(q)
Thursday 30.11.2017	Lecture	Theta functions 1
Tuesday 05.12.2017	Lecture	Theta functions 2
Thursday 07.12.2017	Lecture	Representation of integers by quadratic forms, intro to the circle method after Kloosterman
Tuesday 12.12.2017	Exercise	Exercises. Solutions.
Thursday 14.12.2017	Lecture	The circle method continued
Tuesday 19.12.2017	Exercise	Exercises. Solutions.
Thursday 21.12.2017	Lecture	The exciting conclusion

The new exercises will be posted here on Fridays. We expect you to look at the problems and to prepare questions for the exercise class on Tuesday.

Running list of exercises and solutions

The main resource for the class is the lecture notes posted above. These are based in large part on a course of Prof. Dr. Michel given at EPFL 2011-2013. The lecture also draws on material from the following books:

Topics in Classical Automorphic Forms by H. Iwaniec (AMS 1997)
A First Course in Modular Forms by F. Diamond and J. Shurman (Springer 2005)

The exam will be an oral exam and will last 30 minutes, in which you will be asked to answer questions and solve problems on the blackboard.

Classical Modular Forms Autumn 2017

Exam

Content

Prerequisites

Schedule

Exercises

Literature

Examination