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.

Lesson 19 treats the local quaternion correspondence and its global form over ℚ, together with orbital integrals and characters for division matrix groups. The general higher-degree local and global correspondences are stated without full proofs. Lesson 20 develops dihedral forms and examples; the general purity theorem, the weight-one case and the Kim–Sarnak bound are stated without full proofs.

1. Adèles for GL₂ and strong approximation
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
10. Supercuspidal representations from compact induction
11. Real and complex representations: weights, gamma factors and classical forms
12. Restricted tensor products and the tensor product theorem
13. Unramified representations of GL₂(F) and Satake parameters
14. Global Whittaker functions and the L-function of a cuspidal representation
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

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