Modular Functions and Modular Forms - J.S. Milne   Top
Course Notes
Group Theory
Fields and Galois Theory
Algebraic Geometry
Algebraic Number Theory
Modular Functions and Modular Forms
Elliptic Curves -- see books.
Abelian Varieties
Lectures on Etale Cohomology
Class Field Theory
Complex Multiplication
Basic Theory of Affine Group Schemes
Lie Algebras, Algebraic Groups, and Lie Groups
Reductive Groups

pdf current version (1.31)


This is an introduction to the arithmetic theory of modular functions and modular forms, with a greater emphasis on the geometry than most accounts.


  1. Preliminaries
  2. Elliptic modular curves as Riemann surfaces
  3. Elliptic functions
  4. Modular functions and modular forms
  5. Hecke operators
  6. The modular equation for Gamma_0(N)
  7. The canonical model of X_0(N) over Q
  8. Modular curves as moduli varieties
  9. Modular forms, Dirichlet series, and functional equations
  10. Correspondences on curves; the theorem of Eichler and Shimura
  11. Curves and their zeta functions
  12. Complex multiplication for elliptic curves


The algebra and complex analysis usually covered in advanced undergraduate or first-year graduate courses.


v1.10; May 22, 1997; first version on the web; 128 pages. old version (1.10)
v1.20; November 23, 2009; new style; minor fixes and improvements; addedlist of symbols; 129 pages.
v1.30; April 26, 2012; corrected; many minor revisions; 138 pages.
v1.31; March 22, 2017; corrected; minor revisions; 134 pages.