FSF3633 Modular forms 7.5 credits
This course is aimed at a general PhD level audience and should be accessible and of interest to doctoral students from different directions, including analytic as well as algebraic or geometric directions. Modular forms are a beautiful and central topic in number theory which proved to be a powerful tool for a wide range of applications. The theory of modular forms itself is very broad and combines perspectives from analytic number theory, algebraic geometry as well as representation theory. Of the wide variety of applications the most prominent is certainly the proof of Fermat's Last Theorem. Other applications include explicit constructions of families of Ramanujan graphs as well as an unconditional proof of Linnik's theorem about the equidistribution of lattice points on spheres. The topic of modular forms being so broad means that we will only be able to study some of its aspects. This course will start from a classical point of view and aims at discussing at least one application such as Duke's work on Linnik's problem concerning the distribution of points { n^{-1/2}(x,y,z) | x,y,z ∈ ℤ, x²+y²+z² = n } on the unit sphere in ℝ³ as n→∞, or questions concerning the asymptotic behaviour of the function r_k(n) that counts the number of representations of n as the sum of k perfect squares.
Content and learning outcomes
Course disposition
Lectures, homework.
Course contents
This course is an introduction to the theory of modular forms, starting from a classical point of view. It will discuss topics such as the modular group, modular functions, modular forms, Hecke operators, Dirichlet series, Theta functions, L-functions. Possible additional topics are aspects of the spectral theory of automorphic forms, applications within number theory, or a further study of algebro-geometric aspects.
Intended learning outcomes
After completion of the course the students
- have gained a background in modular forms starting from which the study can be continued in any of the possible directions
- understand and are able to apply techniques from the classical theory of modular forms
Literature and preparations
Specific prerequisites
Completed course in Complex Analysis.
Recommended prerequisites
Completed course in Galois theory
Equipment
Literature
* D. Bump, Automorphic Forms and Representations, Cambridge University Press, 1998
* F. Diamond and J. Shurman, A first course in modular forms, Springer, 2005
* H. Iwaniec, Spectral Methods of Automorphic Forms, American Mathematical Society, 2002
* J.S. Milne, Modular Functions and Modular Forms, Course notes available at www.jmilne.org/math/CourseNotes/mf.html
* P. Sarnak, Some Applications of Modular Forms, Cambridge University Press, 1990
* J-P. Serre, A Course in Arithmetic, Springer-Verlag New York, 1973.
Examination and completion
If the course is discontinued, students may request to be examined during the following two academic years.
Grading scale
Examination
- INL1 - Hand-in assignment, 7.5 credits, grading scale: P, F
Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.
The examiner may apply another examination format when re-examining individual students.
Opportunity to complete the requirements via supplementary examination
Opportunity to raise an approved grade via renewed examination
Examiner
Ethical approach
- All members of a group are responsible for the group's work.
- In any assessment, every student shall honestly disclose any help received and sources used.
- In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.