Modular forms

Modular forms from the geometry of the upper half-plane: modular curves, holomorphic forms, Hecke operators and newforms, L-functions, theta series, cohomology and modular symbols, complex elliptic curves, non-holomorphic Eisenstein series and the Rankin–Selberg method.

1. The upper half-plane and the modular group
2. Congruence subgroups, cusps and elliptic points
3. Modular curves and their genus
4. Modular forms, lattice functions and Eisenstein series
5. The valence formula and the ring of modular forms of level one
6. Dimension formulas for congruence subgroups
7. The Petersson inner product and Poincaré series
8. Hecke operators of level one
9. Hecke operators for Γ_0(N) and Γ_1(N)
10. Oldforms, newforms and the theory of Atkin, Lehner and Li
11. The L-function of a cusp form
12. Twists, level N and Hecke's converse theorem
13. Theta functions and sums of squares
14. Theta series of lattices
15. Group cohomology of Γ and the Eichler–Shimura isomorphism
16. Modular symbols and the algebraicity of Hecke eigenvalues
17. Complex tori, elliptic curves and the moduli interpretation of modular curves
18. Non-holomorphic Eisenstein series and Maass forms
19. The Rankin–Selberg method

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