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 r
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
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
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 no
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 po
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
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
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
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 cohom
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 i
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 eit
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, wri
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 the
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 a
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 ni
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