Automorphic forms and representations of GL(2)

A course on how congruence conditions, Fourier coefficients and local representations fit together for GL(2). It develops the passage from classical modular forms to adelic representations, local and global Whittaker models and L-functions, newvectors, the cuspidal and Eisenstein spectra, and quaternionic forms and the trace formula.

1. Adèles for GL₂ and strong approximation (in preparation)
2. From modular forms to adelic functions
3. Automorphic forms, modules and Hilbert constituents
4. Why the cuspidal spectrum is discrete
5. Smooth local representations and the Hecke-module dictionary
6. Normalized induction and Jacquet modules
7. Whittaker models, Kirillov models and the local classification
8. Newvectors, conductors and the passage to classical newforms
9. Local L-factors, epsilon factors and the local converse theorem (in preparation)
10. Supercuspidal representations from compact induction
11. Real and complex representations: weights, gamma factors and classical forms (in preparation)
12. Restricted tensor products and the tensor product theorem (in preparation)
13. Unramified representations of GL₂(F) and Satake parameters
14. Global Whittaker functions and the L-function of a cuspidal representation (in preparation)
15. Multiplicity one, strong multiplicity one and the converse theorem
16. Eisenstein series on GL₂(A) and the continuous spectrum
17. Quaternion algebras and their automorphic forms
18. The trace formula for a compact quotient
19. The Jacquet–Langlands correspondence
20. Dihedral forms, examples, and the Ramanujan conjecture for GL₂ (in preparation)

Original text CC0 1.0 unless the lesson states other terms.