MA 162B LECTURE NOTES: THURSDAY, MARCH 4
1. Complex Galois Representations
Fix a continuous representation : GQ GLn (C) and an odd prime . We
ker
x some notation: Denote L = Q
as the eld xed by the representation and
ker
e
F =Q
as the eld xed by its pro
MA 162B LECTURE NOTES: TUESDAY, MARCH 2
1. Abelian Varieties of GL2 -Type (contd)
1.1. Absolute Irreducibility (contd). In the previous lecture we explained why
if E is semistable at 3 and 5 then E is modular. We generalize this concept for
abelian variet
MA 162B LECTURE NOTES: THURSDAY, FEBRUARY 26
1. Abelian Varieties of GL2 -Type
1.1. Modularity Criteria. Heres what weve shown so far: Fix a continuous
residual representation : GQ GL(V ), where V is a 2-dimensional vector space
over a nite eld k of odd c
MA 162B LECTURE NOTES: TUESDAY, FEBRUARY 24
1. Proof of Modularity (contd)
1.1. Step #3: Construction of Hecke Modules. Recall that we want to con(2)
struct a family of T -modules L , for nite sets not including , satisfying the
following axioms:
(2)
HM1:
MA 162B LECTURE NOTES: FRIDAY, FEBRUARY 20
1. Proof of Modularity (contd)
1.1. Step #1: The Minimal Case (contd). Recall that in the previous lecture
1
we dened r = dimk H Q, ad0 (1) , and for each positive integer n we have a
(2)
set Q be a set of r prim
MA 162B LECTURE NOTES: THURSDAY, FEBRUARY 19
1. Criteria for Modularity (contd)
1.1. Review of Criteria. Lets review the conditions weve assumed so far. We x
a continuous residual representation : GQ GL(V ), where V is a 2-dimensional
vector space over a
MA 162B LECTURE NOTES: THURSDAY, FEBRUARY 12
1. Modular Deformation Ring (contd)
1.1. Modular Galois Representations (contd). Associated to each ordinary
normalized -adic eigenform F (X ; ) = n an (X ) q n of level N and nebentype
there is a continuous -
MA 162B LECTURE NOTES: TUESDAY, FEBRUARY 10
1. Motivation for -adic Galois Representations
1.1. 1-Dimensional -adic Galois Representations. We recall a few facts in
order to motivate a larger denition.
We focus on det . Say that we have a 2-dimensional co
MA 162B LECTURE NOTES: THURSDAY, FEBRUARY 5
1. Selmer Groups and Deformation Problems (contd)
We now translate the condition of being an innitesimal deformation of type
into local conditions on cohomology classes.
1.1. Deformation Conditions at = . Each
MA 162B LECTURE NOTES: TUESDAY, FEBRUARY 3
1. Universal Deformation Ring (contd)
1.1. Witt Vectors (contd). We call W (k ) the ring of Witt vectors. Note that
by construction W (F ) Z . Sometimes it is helpful to consider the vectors Wn (k )
of nite lengt
MA 162B LECTURE NOTES: FRIDAY, JANUARY 30
1. Examples of Cohomology Groups (contd)
1.1. H 2 and Projective Galois Representations. Say we have a projective
Galois representation : G P GL(V ) where either G = GQ or G = Gp i.e. either
is a local or global
MA 162B LECTURE NOTES: THURSDAY, JANUARY 29
1. Galois Cohomology
1.1. Denitions. Fix a nonnegative integer n, and consider the n-fold product
G G G of a pronite group G. Let X be an abelian group, written additively,
upon which G acts continuously i.e. X
MA 162B LECTURE NOTES: THURSDAY, JANUARY 22
1. Examples: -adic Representations (contd)
1.1. Modular Forms (contd). We focus on cusp forms f ( ) of weight = 2. Recall from the previous lecture that there is a canonical isomorphism from S2 (1 (N )
to those
MA 162B LECTURE NOTES: TUESDAY, JANUARY 20
1. Examples: -adic Representations (contd)
1.1. Construction via Weil Pairing. We sketch a proof of how to construct the
-adic representation associated to an abelian variety. We only discuss the case of
an ellip
MA 162B LECTURE NOTES: THURSDAY, JANUARY 15
1. Examples of Galois Representations: Complex Representations
1.1. Regular Representation. Consider a complex representation
:
Gal Q/Q GLd (C)
ker
with nite image. If we denote L = Q
then we have a faithful ma
MA 162B LECTURE NOTES: TUESDAY, JANUARY 13
1. What is a Galois Representation?
1.1. Basic Denitions. Let V be a nite dimensional vector space over a eld K .
Any continuous homomorphism
:
Gal Q/Q GL(V )
is called a Galois representation. There are three ty
MA 162B LECTURE NOTES: TUESDAY, JANUARY 6
1. Course Information
1.1. Instructor. Edray Goins. Oce: 276 Sloan. Extension: x4347. E-Mail:
goins@caltech.edu.
1.2. Meeting Times. Tuesdays and Thursdays from 11:00 AM through 12:30 PM
in 153 Sloan. Oce hours wi