COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 M. FLACH 1. Global class field theory in classical terms Recall the isomorphism Z Z × −!∼ Q Q 7! 7! a ( =m ) Gal( (ζm)= ); a (ζm ζm) where ζm is a primitive m-th root of unity. This is the reciprocity isomorphism of global class field theory for the number field Q. What could a generalization of this to any number field K be? Of course one has a natural map (going the other way) ∼ × Gal(K(ζm)=K) ! Aut(µm) = (Z=mZ) ; σ 7! (ζ 7! σ(ζ)) called the cyclotomic character. It is even an isomorphism if K is linearly disjoint from Q(ζm). But that's not class field theory, or rather it's only a tiny part of class field theory for the field K. The proper, and much more stunning generalisation is a reciprocity isomorphism ∼ Hm −! Gal(K(m)=K) between so called ray class groups Hm of modulus m and Galois groups of corre- sponding finite abelian extensions K(m)=K. In a first approximation one can think of a modulus m as an integral ideal in OK but it is slightly more general. The ray m × class group H isn't just (OK =m) although the two groups are somewhat related. In any case Hm is explicit, or one should say easily computable as soon as one can O× O compute the unit group K and the class group Cl( K ) of the field K. Note that both are trivial for K = Q, so that ray class groups are particularly easy for K = Q. So the first part of class field theory states the existence of this family of abelian extensions K(m)=K with explicit Galois groups. Then there is a complementary statement saying that any abelian extension L=K is contained in some ray class field and one can pin down which in terms of the ramification of L=K. For K = Q this is the Kronecker Weber theorem which can be proved directly (independently of CFT) using some ramification theory (it's a sequence of exercises in Marcus' book). 1.1. Places. In order to rigorously define what a modulus m is, we talk about the concept of places. Recall that OK is a Dedekind ring and any integral ideal m has a unique factorization Y m = pnp p ≥0 into a product of prime ideals where np 2 Z and almost all np = 0. We have to add the ”infinite primes" to define a general modulus. Definition 1.1.1. For a field K a valuation is a function j−j : K ! R≥0 satisfying 1 2 M. FLACH (i) jaj = 0 if and only if a = 0. (ii) jabj = jaj · jbj. (iii) There is a constant C so that ja + 1j ≤ C whenever jaj ≤ 1. Remark 1.1.1. We leave it as an exercise to show that if one can take C = 2 in (iii) then j − j satisfies the triangle inequality ja + bj ≤ jaj + jbj and if one can take C = 1 then it satisfies the even stronger inequality ja + bj ≤ maxfjaj; jbjg: Valuations of this last type are called non-archimedean. Example 1.1.1. The trivial valuation is the valuation given by jaj = 1 for all a =6 0. Example 1.1.2. If ι : K ! L is an embedding of K into a local field , then the valuation of L induces one on K. In particular any prime ideal p of OK induces ∼ r1 r2 a valuation via K ! Kp. If L is a direct factor of R ⊗Q K = R × C then the usual archimedean absolute value on R or C induces one on K. Definition 1.1.2. Two valuations j − j1 and j − j2 are equivalent if there is c > 0 j j j jc 2 with a 1 = a 2 for all a K. Lemma 1.1. Two valuations j − j1 and j − j2 are equivalent if and only if 8α 2 K jαj1 < 1 , jαj2 < 1: Proof. One direction is clear. Let y 2 K be such that jyj1 > 1 which exists since × j j1 is nontrivial. We show that we can take c = log jyj1= log jyj2. For any x 2 K there is b 2 R with j j j jb x 1 = y 1: Approximating b by rational numbers b < mi=ni from above we have m =n j j j jb j j i i ) j ni mi j x 1 = y 1 < y 1 x =y 1 < 1 which implies m =n j ni mi j ) j j j j i i x =y 2 < 1 x 2 < y 2 and hence j j ≤ j jb x 2 y 2 by passing to the limit. Using a sequence approximating b from below we obtain j j ≥ j jb j j j jb x 2 y 2 and hence x 2 = y 2. So j j j jb j jc b j jb c j jc x 1 = y 1 = ( y 2) = ( y 2) = x 2: The main result about valuations on general fields is: Theorem 1.1. (Weak approximation) If j j1;:::; j jN are inequivalent valuations on a field K (not necessarily global), α1; : : : ; αN 2 K, ϵ > 0 then there exists ξ 2 K with jαn − ξjn < ϵ for 1 ≤ n ≤ N. COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 3 Remark 1.1.2. If K is a global field and all j jn are non-archimedean, then this is basically the Chinese remainder theorem. Take a Dedekind ring O ⊆ K so that the j jn correspond to prime ideals pn of O, multiply the αn by a common denominator in Z to make them integral (this will rescale ϵ by a positive factor) then take en −en large enough so that Npn < ϵ. Then by the Chinese remainder theorem there − 2 O ≡ en j − j ≤ en exists ξ , ξ αn mod pn , i.e. αn ξ n Npn < ϵ. Proof. We note first that it will be enough to find θn 2 K with jθnjn > 1; jθnjm < 1 n =6 m: Then we have ( θr 1 w.r.t. j j lim n = n !1 r r 1 + θn 0 w.r.t. j jm, m =6 n and XN θr ξ = n · α 1 + θr n n=1 n will have the required property for r sufficiently large. By symmetry it is enough to show existence of θ = θ1 with the above property. This we do by induction on N. For N = 2 by Lemma 1.1 we find α and β with jαj1 < 1; jαj2 ≥ 1; jβj1 ≥ 1; jβj2 < 1 and can take θ = β/α. By the induction assumption there is ϕ 2 K with jϕj1 > 1; jϕjn < 1; 2 ≤ n ≤ N − 1 and 2 K with j j1 > 1; j jN < 1: Then set 8 <>ϕ if jϕjN < 1 r θ = ϕ if jϕjN = 1 :> r ϕ j j 1+ϕr if ϕ N > 1 where r 2 Z is sufficiently large. Proposition 1.1. If K is a global field there are natural bijections between a) The equivalence classes of nontrivial valuations on K. b) The equivalence classes of dense embeddings ι : K ! L where L is a locally compact field and two embeddings ι, ι0 are equivalent if there is a continuous ∼ 0 0 isomorphism ϕ : L = L so that ι = ϕ ◦ ι. c) If K is a number field, 2 fpjf0g 6= p ⊂ OK primeg [ fe 2 R ⊗Q Kje = e; e indecomposableg Proof. (sketch). By restriction one obtains a valuation on the subfield Q or Fp(T ). These are classified by Ostrowski's theorem which we won't prove. On Q they are the p-adic valuations j − j and the usual absolute value j − j = j − j1. Then ∼ p Qp ⊗Q K = K1 ×· · ·×Kr and each Ki is a finite extension of Qp, hence has a unique extension j − ji of j − jp and the Ki are complete. The given valuation on K induces a function on Qp ⊗Q K by continuity. This function is nonzero (since nonzero on Qp) so must be nonzero on one of the Ki and is a valuation there, so is equivalent to j − ji. Since K is dense in Qp ⊗Q K an element (0;:::;y;:::; 0) corresponding to 4 M. FLACH y 2 Ki, jyji > 1 can be approximated arbitrarily close by an element in K which shows that j − ji is independent from the other valuations. Finally, one can classify locally compact fields (i.e. fields with a locally compact topology so that addition, multiplication and inversion are continuous). Any such field L is isomorphic to either R, C, a finite extension of Qp or a Laurent series field Fq((T )), in other words to a completion of a global field at some valuation. Definition 1.1.3. A place of a global field K is an element of the set in Proposition 1.1. A place is called non-archimedean (or finite), archimedean (or infinite), real, or complex if the corresponding local field is. We denote places by p, the corresponding embeddings by ιp and, if p is finite, the corresponding discrete valuations by vp. Definition 1.1.4. Define normalized valuations j − j on a global field by 8 p − <>Np vp(x) p finite j j j j x p = > ιp(x) p real : 2 jιp(x)j p complex. They satisfy the product formula Y jαjp = 1: all places p 1.2.