Class Field Theory and the Local-Global Property Noah Taylor Discussed with Professor Frank Calegari March 18, 2017 1 Classical Class Field Theory 1.1 Lubin Tate Theory In this section we derive a concrete description of a relationship between the Galois extensions of a local field K with abelian Galois groups and the open subgroups of K×. We assume that K is a finite extension of Qp, OK has maximal ideal m, and we suppose the residue field k of K has size q, a power of p. Lemma 1.1 (Hensel). If f(x) is a polynomial in O [x] and a 2 O with f(a) < 1 then there is K K f 0(a)2 some a with f(a ) = 0 and ja − a j ≤ f(a) . 1 1 1 f 0(a) f(a) Proof. As in Newton's method, we continually replace a with a− f 0(a) . Careful consideration of the valuation of f(a) shows that it decreases while f 0(a) does not, so that f(a) tends to 0. Because K is complete, the sequence of a's we get has a limit a1, and we find that f(a1) = 0. The statement about the distance between a and a1 follows immediately. We can apply this first to the polynomial f(x) = xq − x and a being any representative of any element in k to find that K contains the q − 1'st roots of unity. It also follows that if L=K is an unramified extension of degree n, then L = K(ζ) for a qn − 1'st root of unity ζ. In particular, L=K is an abelian extension. Proposition 1.2. If L=K is unramified, then Gal(L=K) is cyclic, generated by Frob : ζ ! ζq. q More generally, there is a unique generator Frob for which Frob(x) ≡ x mod m for all x 2 OL. Proposition 1.3. The compositum of two abelian extensions of K is another abelian extension of K; the compositum of two unramified extensions of K is another unramified extension of K. So we may talk about Kab=K and Kur=K, the maximal abelian and unramified extensions of K. We let π be any uniformizer of K. Lemma 1.4. Given a polynomial f(x) = xq + πx2u(x) + πx for some u, there is a unique power series F 2 OK x; y with F (x; y) = F (y; x), F (x; 0) = x, F (F (x; y); z) = F (x; F (y; z)) and J K f(F (x; y)) = F (f(x); f(y)). Furthermore, for a 2 OK , there is a unique [a] 2 OK x with [a] ≡ ax mod (x2) and [a] ◦ f = f ◦ [a]. J K Proof. [1] I.2.11. 1 Theorem 1.5. Suppose f is defined as above. Further suppose Λn;f = fx : f ◦ f ◦ ::: ◦ f(x) = 0g, | {zn } 1 and Kn;f = K(Λn;f ), and Kf = [n=1Kn;f . Then Kf =K is a totally ramified abelian extension, ab ur ur ab ur K = K · Kf and Kf \ K = K. Thus Gal(K ) = Gal(Kf ) × Gal(K ). f qn−1(q−1) Proof. The polynomial x ◦ f ◦ ::: ◦ f can be written as π + x · π(:::) + x , so it is irreducible by Eisenstein and the roots, Λn;f nΛn−1;f , are all conjugates. Noting that F ([a]; [b]) ≡ (a + b)x mod (x2) and F ([a]; [b]) ◦ f = F (f ◦ [a]; f ◦ [b]) = f ◦ F ([a]; [b]), and remembering uniqueness, we must have F ([a]; [b]) = [a+b]. Similarly, [a]◦[b] = [ab]. Now [a](Λn;f ) ⊆ Λn;f because [a] commutes k n with f, and since f ◦ f ◦ ::: ◦ f = [π ], we see that [a](Λn;f ) = 0 if and only if π ja. Then this | {z } k n gives an action of OK =(π ) on Λn;f . If x 2 Λn;f nΛn−1;f , then the action sends x everywhere. And since [b]([a](x)) = [ab](x), as long as π - a, x ! [a](x) is an automorphism of Λn;f and extends to n × an automorphism of Kn;f . So Gal(Kn;f =K) ' (OK =(π )) , and is thus abelian. It's easy to check × × that these are compatible for different n and extend to an isomorphism Kf ' Od = O . The other parts are in [1] I.1-3. Proposition 1.6. The fields Kn;f depend not on f but only on the uniformizer π. As long as 2 q f ≡ πx mod (x ) and ≡ x mod π, different f's give the same fields Kn;f . Thus we write Kn,π and Kπ instead. And [a], while a priori different symbols for different f, gives the same action on Kπ for different f's. Theorem 1.7. There is a map φ from K× to Gal(Kab=K) which takes uniformizers π to Frob ur × × when restricted to K and, given an abelian extension L=K, can be restricted to K =NL=K (L ) ' × Gal(L=K). That is, if we restrict φ to Gal(L=K), then the kernel is exactly NL=K (L ), the norm subgroup of L over K. Proof. The map is given by k −1 k ur ab x = u · π ! ([u ]; Frob ) 2 Gal(Kπ) × Gal(K ) = Gal(K ): This is in fact invariant for different π's. For proof of this theorem and the previous (nonobvious) proposition, see [1] I.3. × × Theorem 1.8. Any finite-index open subgroup of K is equal to the norm group NL=K (L ) of a unique finite abelian extension L=K. p Example 1.9 (Local Kronecker-Weber). If K = Qp, we can take π = p, f(x) = (x + 1) − 1, F (x; y) = (x+1)(y +1)−1, and [u] = (x+1)u −1, where this exponent is defined because of Lucas' i Theorem. This gives Λn;f = fµpn −1g, and any abelian extension of Qp is contained in a cyclotomic ( ( ζ ! ζp ζ ! ζ extension. The corresponding φ takes p to m m and x 2 O× to m m . K x−1 ζpn ! ζpn ζpn ! ζpn Remark 1.10. The key point is not really the construction of φ, although that was a major triumph of Tate and Lubin. The main draw of local class field theory is that it describes a certain class of representations of Gal(K=K), namely the one-dimensional ones. It shows that they have this very nice structure: if we complete K× with respect to its finite-dimensional quotients to get Kd×, this is ab × isomorphic to Gal(K =K). And Kd has a nice describable structure, namely Zb × µK × (1 + πOK ). So we get a description of all 1-dimensional representations of Gal(K=K), an extrinsic property of K depending on its algebraic closure, depending very explicitly on Kd× and hence on only K×. 2 1.2 Global Class Field Theory Let K be a global field. We write Kv to be the completion of K with respect to the absolute value given by v. Q0 × Definition 1.11. The ideles of K are defined as IK = v Kv over all places v, where the product × is restricted so that for all but finitely many v, the element is in Ov . A basis for the topology is Q0 × × given by v Uv, Uv open in Kv , and the product again restricted so that Un is almost always Ov . × × The diagonal embedding of K into IK is discrete, so we can define CK = IK =K , the idele class group. This will play the same role as K× did in local class field theory. For a finite extension L=K, we can embed IK into IL: the coordinate of w, a place over v, is av. Q We can go back the other way as well: the norm of an idele (aw)w in IL is ( wjv NLw=Kv (aw))v. × × Note that NL=K takes the embedding of L in IL to the embedding of K in IK , so we can define the norm map from CL to CK . ab Theorem 1.12. There is a homomorphism φ : CK ! Gal(K =K) which can be restricted to an isomorphism CK =NL=K (CL) ! Gal(L=K) for Galois extensions L=K. Proof. We first define a map φL=K : IK ! Gal(L=K) for some abelian extension L by tak- Q ing φL=K ((av)v) = v φv(av), where φv is the local map defined above, but restricted from ab Gal(Kv =Kv) to Gal(Lw=Kv) to Gal(L=K). (If v is real infinite, φv sends positive numbers to the identity and negative numbers to conjugation in Gal(C=R); if v is complex infinite then there is no Galois group.) This map is well defined, because all but finitely many v are unramified in the extension, meaning that they only act as powers of Frob; and all but finitely many of those v have a 2 O× , so it is the zeroth power of Frob, or the identity. Thus the product is actually finite. v Kv Notice that NL=K (IL) is in the kernel, because all Lw are isomorphic for all w over v since L=K is Galois. So for any v and the section of IL,(awi )wijv, the product of their norms in Kv maps to the identity in the Galois group Gal(Lw=Kv) because each norm separately does, by definition. 0 This φL=K is functorial in L (that is, φL=K jL0 = φL0=K if K ⊂ L ⊂ L) by the local Artin map's ab own functoriality; we can thus define φ : CK ! Gal(K =K) by inverse limit.