Math 847 : Notes April 9
Lalit Jain
May 1, 2013
1
Recap
Say : GQ GL2 (F ) is an asbolutely irreducible, semistable representation
associated to a cuspidal eigenform f. We are interested in lifts of type of to
complete Noetherian local rings with residue e
1
Strategy
In the summer of 1986, Ken Ribet succeeded in proving the epsilon conjecture, which was (at least partially) a proof
(Weak Serre Conjecture) (Strong Serre Conjecture)
Let E be the Frey curve for a contradiction to F.L.T. and E, the associated
r
Fermats Last Theorem
Math 847 - March 7, 2013
We dene semistable elliptic curves and show using Tate elliptic curves that the corresponding Galois
representations are semistable.
Recall the Frey elliptic curve y 2 = x(x A)(x + B ), A + B + C = 0, where A
Notes for March 5, 2013
Recall for a cuspidal eigenform f of type (k, N, ), a prime ideal of Of such that K = Frac(Of ) )
and Z = lZ , we obtained a modular Galois representation : GQ GL2 (K ) which is unramied
at p lN and
Tr(Frobp ) = ap (f )
det (Frobp
Notes for March 14, 2013
Let us x : GQ
GQ,S
and consider
/ GL2 (Fl )
s99
ss
ss
ss
ss
cfw_semistable lifts of
cfw_ semistable modular lifts of
cfw_semistable elliptic lifts of
We ask if
cfw_ semistable modular lifts of
=
cfw_semistable lifts of
hold.
MATH 847 - TOPICS IN ALGEBRA COURSE NOTES: A PROOF OF FERMATS
LAST THEOREM
SPRING 2013
Notes for March 19, 2013
by Jonathan Lima
Set-Up. Fix a nite eld F, such that char(F) = l > 2. Fix a continuous homomorphism : GQ GL2 (F).
Suppose that is absolutely re
Fermats Last Theorem
Math 847 - April 4, 2013
Let : GQ GL2 (F ) be an absolutely irreducible, semistable, continuous representation associated
to a modular form f , of type . (F is a nite eld with charF = l > 2). Let be a nite set of
rational primes. We h
Notes for Apr 2, 2013
1
Review
Say K is the unique unramied extension of Ql , of degree r. We have
GQl /Il Z
GQl
W (Flr ) K
=
,
Zl
Ql
Z/r
Assume : GQ GL2 (F) an absolutely irreducible and semistable Galois representation, where
F is a nite eld with chara
Fermats Last Theorem
Math 847 - March 21, 2013
Set-Up: Start with continuous : GQ GL2 (F ) with F a nite eld of characteristic l > 2,
absolutely irreducible and semistable, and a nite set of rational primes. Consider the bijection
H 1 (GQ , Ad() cfw_Defo
First, A Reference
Do these notes not make sense? Then you should probably read Modular Forms and Modular Curves by
Diamond and Im.
The Big Picture
Let GQ be the absolute Galois group of Q. We know that X0 (N ) is an algebraic curve over Q. The last time
Fermats Last Theorem
Math 847 - February 26, 2013
Basic Facts on Riemann Surfaces
Denition:
Let cfw_U , and cfw_V , be atlases for R and S respectively. Call a continuous map
f : R S (R, S Riemann surfaces) analytic if the maps f 1 are analytic
maps on
January 24
Suppose that we have a counterexample to (FLT)p of the form ap + bp = cp . Consider the associated Frey curve E : y 2 = x(x ap )(x + bp ) and the associated Galois
representations E,n : GQ GL2 (Z/n).
Conjecture (Taniyama 1955, Shimura). Given a
0.1
Pronite Groups (continued)
Denition 0.1.1. lim Gi , the inverse limit of Gi is the terminal object in the new category. That
is, lim Gi is the unique object (X, cfw_i ) such that there exists exactly one homomorphism to it
from any other object.
Theor
Math 847 - Topics in Algebra Course Notes: A Proof
of Fermats Last Theorem
Spring 2013
January 26, 2013
Chapter 1
Background and History
1.1
Pythagorean triples
Consider Pythagorean triples (x, y, z ) so that x2 + y 2 = z 2 . For example, 32 + 42 = 52 . T
MATH 847 - TOPICS IN ALGEBRA COURSE NOTES: A PROOF OF FERMATS
LAST THEOREM
SPRING 2013
Notes for February 5, 2013
by Vladimir Sotirov
1.1. Recap from last time. For L/K a (possibly innite) separable algebraic Galois extension we have
L=
i
Li
Li
cfw_Li /K
847 Notes February 6 2013
1
Valuations on extensions of the p-adics
Denition 1.1. Let K/Qp be a nite Galois extension. We can dene a Norm on K via the map N :
K * Q* , N (x) = Gal(K/Qp ) (x). Given this Norm, we may dene a discrete valuation on K via the
Fermats Last Theorem
Math 847 - February 21, 2013
We continue to explore the relationships between various collections of Galois representations. We
have the inclusion
cfw_modular Galois representations cfw_nice Galois representations,
which wed like to s
Fermats Last Theorem
Math 847 - February 19, 2013
What is an elliptic curve over Q?
Its rather unsatisfactory to say E : y 2 = f (x), f cubic with coecients in Q and distinct
root in Q. What we can instead say is an elliptic curve over Q is a bunch of gro
847 Notes
14 February 2014
From last time: suppose M L. We were left to prove the commutativity of the following
(L)
/
( L )
L
/ k
L
(L)
G0 /G1
(M )
(M )
G0 /G1
(mod mL )
?
/ k
M
/
(M )
M
(mod mM )
The question mark is the Norm function.
Galois Represen
Notes, February 12
In the following we let G = Gal(K/Qp ), where K/Qp is a nite Galois extension. As usual we have a
discrete valuation w on K s.t. w( ) = 1, k = A/m, where is the uniformizer, and A = cfw_x K |w(x) 0,
m = cfw_x K |w(x) > 0.
Recall that Gi