Math 847 - Topics in Algebra Course Notes: a Proof of Fermat’S Last Theorem Spring 2013

MATH 847 - TOPICS IN ALGEBRA COURSE NOTES: A PROOF OF FERMAT’S LAST THEOREM SPRING 2013

Notes for February 5, 2013 by Vladimir Sotirov 1.1. Recap from last time. For L/K a (possibly inﬁnite) separable algebraic Galois extension we have S L = i Li

··· Li ··· {Li/K} = ﬁnite Galois extensions

K ∼ Q Furthermore, Gal(L/K) = lim Gal(Li/K) ≤ i Gal(Li/K) where the inverse limit lim Gal(Li/K) consists ←−Q ←− of those elements of the product i Gal(Li/K) that are compatible (with the inclusions of fields). If we give the finite groups Gal(L /K) the discrete topology, then lim Gal(L /K) is a profinite group and i ←− i is compact, Hausdorff and totally disconnected. We will be interested in the continuous homomorphisms out of it. In fact, there is an order-reversing bijection: intermediate fields←→closed subgroups of Gal(L/K). Remark (Notation). Gal(K/K) = G . For example, G =∼ = lim /n. K Fp Zb ←− Z n Note that Z ≤ Zb is not a closed subgroup (since its closure is the whole thing). In particular, Z does not correspond to any intermediate subfield. (aside: let K =compositum of all quadratic extensions of Q. Then Q Gal(K/Q) =∼ Z/2, and the diagonal subgroups are not closed, do not correspond to any extension) 1.2. Local rings continued. Throughout R is a commutative ring with 1, and I is an ideal. Recall that R = lim R/Ii is called the I-adic completion of R. For example, when R = , I = p , we I ←− Z Z have RI = Zp = p-adic integers=p-adic completion of Z).

Exercise. If I is maximal, then RI is local (i.e. has a unique maximal ideal).

We will be interested in (families of) continuous homomorphisms GQ → GL2(Zp).

Example (Hida, 1983). Hida found a representation GQ → GL2(Zp[[T ]]) such that the composition with k−1 the ring homomorphism Zp[[T ]] → Zp deﬁned by T 7→ (1 + p) − 1 yields a modular representation for each k ≥ 2. Note that Zp[[T ]] is also a local ring, but its maximal ideal is not principal: it is generated by (p, T ).

We now describe the category C of complete Noetherian local rings with residue ﬁeld Fp (residue ﬁeld=quotient by unique maximal ideal). Its objects have a nice concrete representation due to Cohen:

Theorem (Cohen). All such rings are of the form Zp[[T1,...,Tr]]/I. Morphisms of C are ring homomorphisms φ: R → S such that the following diagram commutes

φ R / S

id R/mR / S/mS

(equivalently: such that φ(mR) ⊂ mS) 1 MATH 847 - TOPICS IN ALGEBRA COURSE NOTES: A PROOF OF FERMAT’S LAST THEOREM SPRING 2013 2

Exercise. If R is an object of C, then GL (R) ∼ lim GL (R/Ii) n = ←− n Note that we can also define the category C in a different way: it is the category of complete Noetherian local rings with designated homomorphisms to Fp. This is equivalent to our original definition because local rings with a ring homomorphism to Fp necessarily have residue field Fp, and because there is exactly one ring homomorphism to Fp from any local ring with residue field Fp. We mention this because in addition to C, we will sometimes also make use of the category of complete Noetherian local rings with designated homomorphisms to Fq for q a prime power. Another theorem of Cohen gives us an explicit description of the rings involved:

Theorem (Cohen). Complete Noetherian local rings with residue field Fq are of the form W (Fq)[[T1,...,Tr]]/I where W (Fq) is the ring of infinite Witt vectors. (W (Fq) is sometimes called Zq by cryptographers) Morphisms in this category will be ring homomorphisms that commute with the designated ring homomorphisms to Fq. Note that this agrees with how we defined the morphisms in C since any local ring R admits at most one homomorphism to Fp. 2. Internal Structure of Infinite Galois Groups

i j We begin with Zp = {(a1, a2,... ): ai ∈ Z/p , ai (mod p ) = aj if i ≥ j}. We have projection maps n n 2 Zp → Z/p which we can check have kernels p Zp. We also have inclusions Zp ⊇ pZp ⊇ p Zp ⊇ · · · .

Definition. We define the p-adic valuation vp : Zp → {0, 1, 2,... } ∪ {+∞} as follows. n n+1 If 0 6= x ∈ p Zp − p Zp, then we set vp(x) = n. For x = 0 we set vp(x) = +∞. T n Since n p Zp = {0}, vp is well-defined (more abstractly: by Krull’s theorem).

Exercise. x is a unit of Zp if and only if vp(x) = 0.

vp(x) Corollary. Every non-zero element x of Zp can be written uniquely as p u where u is a unit.

The map vp has the following properties:

(1) vp(xy) = vp(x) + vp(y) (with addition suitably interpreted if x or y is 0). m n m n−m (2) vp(x + y) ≥ min(vp(x), vp(y)) since p u + p v = p (u + p v) when n ≥ m. ( cvp(x−y) x 6= y Deﬁnition. The p-adic metric on Zp is given by d(x, y) = where 0 < c < 1. By the 0 x = y

properties of vp this satisﬁes the triangle inequality (in fact a stronger version: all triangles are isosceles). n n n An immediate conclusion is that x ∈ p Zp ⇐⇒ vp(x) ≥ n ⇐⇒ d(x, 0) ≤ c . This shows that p Zp ⊂ Zp are closed balls in this topology. Furthermore, Zp is an integral domain by property 1) of vp.

Definition. We define Qp to be the field of fractions of Zp (p-adic numbers). We have the following commutative diagram (by universal properties of fraction fields, and algebraic closures): Q / Qp O O

? ? Q / Qp O O

? ? Z / Zp

By restriction, there is an induced map GQp → GQ.

Exercise. Check that GQp → GQ is a continuous group homomorphism.

Deﬁnition. Given a continuous group homomorphism ρ: GQ → GLn(R) with R an object of C, let ρp : GQp → GQ → GLn(R) denote the composition. Then the collection of ρp (p = 2, 3, 5,... ) is called the local data associated to ρ. MATH 847 - TOPICS IN ALGEBRA COURSE NOTES: A PROOF OF FERMAT’S LAST THEOREM SPRING 2013 3

The point of the local data is that we don’t really understand GQ at all, but we understand the GQp a lot better than GQ (we even have presentations for them when p 6= 2). In particular, we can describe the representations GQp → GLn(R) more easily than the representations GQ → GLn(R). To understand GQp , however, we need to understand Gal(K/Qp) for finite Galois extensions K/Qp. This we do by extending the notion of a valuation to such K as follows. n n Consider Zp − {0} = {p u: n ≥ 0 integer, u a unit}. Then Qp − {0} = {p u: n ∈ Z, u a unit in Zp}. a In this way we extend vp to Qp by setting vp( b ) = vp(a) − vp(b) (where a, b ∈ Zp). So now we have a map vp : Qp → Z ∪ {+∞} with: (1) vp(xy) = vp(x) + vp(y) (suitably interpreted if x or y is 0). (2) vp(x + y) ≥ min(vp(x), vp(y)) × So vp : Qp → Z is a group homomorphism. Definition. A discrete valuation w on a field K is a surjective homomorphism w : K× → Z such that w(x + y) ≥ min(w(x), w(y)) for all x, y ∈ K with w(0) = +∞.

Example. vp on Qp is a discrete valuation. Definition. Given a discrete valuation w on K, set A = {x ∈ K : w(x) ≥ 0} and m = {x ∈ K : w(x) > 0}. We call A a valuation ring; m is its unique maximal ideal (and K is its field of fractions). Exercise. Choose π ∈ K such that w(π) = 1. We will call π a uniformizer. Then m = hπi ⊂ A. Every element x of K× can be written as πw(x)u where u ∈ A − m. Hence in particular A is a PID. Definition. A/m is the residue field of the discrete valuation w on K. × × Definition. Let K/Qp be a finite Galois extension. We have the norm N : K → Qp given by x 7→ v Q σ(x) (this is a group homomorphism). We get a group homomorphism K× →N × →p which σ∈Gal(K/Qp) Qp Z may not be surjective, but its image will be of finite index, say its image is fZ. We define the valuation × vp(N(x)) w : K → Z on K by w(x) = f .