Modular Functions and Modular Forms
(Elliptic Modular Curves)
J.S. Milne
Version 1.30
April 26, 2012
This is an introduction to the arithmetic theory of modular functions and modular forms,
with a gre
Abelian varieties and Jacobians
November 25, 2013
1
Abelian varieties and Jacobians
An abelian variety A over a eld K is an irreducible smooth projective group
variety. In other words, A is a smooth p
Cusp forms and the Eichler-Shimura relation
September 9, 2013
In the last lecture we observed that the family of modular curves X0 (N )
has a model over the rationals. In this lecture we use this fact
Modular Curves
September 4, 2013
The rst examples of Shimura varieties we encounter are the modular
curves. In this lecture we review the basics of modular curves, beginning with
the complex theory an
MA 843 Assignment 2, due Sept. 24
September 13, 2013
1. Let f : X Y be a nite morphism of degree d between smooth projective curves over a eld K .
(a) Show that if D is a divisor on Y of degree d, the
MA 843 Assignment 3, due Oct. 17
October 11, 2013
1. (a) Show that the action of Sp2g (R) on Hg by fractional linear transformations
AB
Z = (A + BZ )(C + DZ )1
CD
is (a) well-dened (in the sense that
Modular abelian varieties and the
Eichler-Shimura relation
October 4, 2013
The goal of this lecture is to construct the Galois representation arising
from a cuspidal eigenform of weight 2.
1
Modular a
Hodge Structures
October 8, 2013
1
A few examples of symmetric spaces
The upper half-plane H is the quotient of SL2 (R) by its maximal compact
subgroup SO(2). More generally, Siegel upper-half space H
The formalism of Shimura varieties
November 25, 2013
1
Shimura varieties: general denition
Denition 1.0.1. A Shimura datum is a pair (G, X ), where G/Q is a reductive group, and X is a G(R)-conjugacy
The Baily-Borel compactication
November 12, 2013
Let G be a simple algebraic group dened over Q, and suppose that
X = G(R)/K is a hermitian symmetric domain. (Note that this already
places restriction
Hermitian Symmetric Domains
November 11, 2013
1
The Deligne torus, and Hodge structures
Let S be the real algebraic group ResC/R Gm . Thus S (R) = C .
If V is a nite-dimensional real vector space, the
Kahler manifolds and variations of Hodge
structures
October 21, 2013
1
Some amazing facts about Khler mania
folds
The best source for this is Claire Voisins wonderful book Hodge Theory and
Complex Alg
Variations of Hodge Structure, part 2
October 28, 2013
Let S be a complex manifold, let f : X S be a family of compact Khler
a
manifolds, and let s S be a base point. Fix an integer k , and let V =
H
MA 843 Assignment 1, due Sept. 12
September 2, 2013
1. Show that there is a homeomorphism SL2 (R)/SO(2) H.
2. If SL2 (Z) is a nite-index subgroup, then \H is a compact Riemann surface minus nitely man