LECTURE NOTES, WEEK 7 MATH 222A, ALGEBRAIC NUMBER THEORY MARTIN H. WEISSMAN Abstract. We discuss the connection between quadratic reciprocity, the Hilbert symbol, and quadratic class ¯eld theory. We also introduce the \adelic per- spective". 1. Quadratic Class Field Theory The main theorems of (local and global) quadratic class ¯eld theory are essen- tially proven in Serre's course in arithmetic. We begin by making this connection explicit and clear. 1.1. Local quadratic class ¯eld theory. Serre proves the following result about the (quadratic) Hilbert symbol over Qp or R: Theorem 1.1. The Hilbert symbol (¢; ¢)v is a non-degenerate bilinear form on the £ £2 F2-vector space Qv =Qv . £ £2 £ Let Sv = Qv =Qv . If x 2 Qv , then we writex ¹ for its image in the vector space Sv. This is a ¯nite-dimensional F2-vector space. Its dimension is 1, 2, or 3, depending on whether v = 1, 2 < v < 1, or v = 2, respectively. It is not a deep theorem that Sv is isomorphic to its dual vector space { this is true for every ¯nite-dimensional vector space, over any ¯eld. The deep theorem (non-degeneracy of the Hilbert symbol) is that the Hilbert symbol \induces" such 0 0 a duality. Let Hv : Sv ! Sv (where Sv is the dual vector space) denote the linear map given by: Hv(a)(b) = (a; b)v: Non-degeneracy means that Hv is an isomorphic of F2-vector spaces; this is what is proven in Serre. The following is the statement of local class ¯eld theory (over Qp or R): Theorem 1.2. Let Gv = Gal(Q¹ v=Qv). Then, if A is any ¯nite abelian group, then there is a group isomorphism: (A) recv : Hom(Gv;A) ! Hom(Qv;A): Of course, to prove this theorem, it su±ces to assume that A is cyclic, by the structure theorem for ¯nite abelian groups. We can use the Hilbert symbol to prove this theorem in the quadratic case: Theorem 1.3. Let Gv = Gal(Q¹ v=Qv). Then there is a group isomorphism: (2) recv : Hom(Gv; Z=2Z) ! Hom(Qv; Z=2Z): Proof. The construction of the \reciprocity" map recv goes as follows: 1 2 MARTIN H. WEISSMAN ² Begin with a nontrivial homomorphism γ : Gv ! Z=2Z. Equivalently, we 0 may begin with a subgroup Gv of index two in Gv. 0 ² Associated to the subgroup Gv of index two, we have a ¯eld extension Kv=Qv of degree two. ² Associated to the ¯eld extension Kv=Qv of degree two, we have the sub- £ £ £2 £ group NKv ½ Qv . Note that Qv ½ NKv p £ £ ² If Kv = Qv( a), with a 2 Qv , then b 2 NKv if and only if (b; a)v = 1 (Chapter III, Section 1, Proposition 1 of Serre). In other words, £ £2 NKv =Qv = ker(Hv(¹a)): By non-degeneracy of the Hilbert symbol, the kernel of the nonzero linear £ £ functional Hv(¹a) has codimension 1. Thus NKv has index 2 in Qv . £ £ ² Associated to the subgroup NKv of index two in Qv , there is a unique £ £ homomorphism recv(γ) from Qv to Z=2Z, with kernel NKv . We construct an inverse to the reciprocity map as follows: ² Suppose that ´ 2 Hom(Qv; Z=2Z) is a nontrivial homomorphism. Equiva- £ lently, we may begin with a subgroup E of Qv of index 2. £2 ² Since E has index two, E ⊃ Qv . £ £2 ² Thus, E is uniquely determined by an index two subgroup of Qv =Qv . Equivalently, E is uniquely determined by a codimension 1 subspace Ov of Sv. ² By the non-degeneracy of the Hilbert symbol, every codimension 1 subspace of Sv arises as the kernel of Hv(¹a) for some 0 6=a ¹ 2 Sv. £ ² Such an elementa ¹ lifts to an element a of Qv , unique up to multiplication £2 by Qv . p ² Thus we get a quadratic ¯eld extension Kv = Qv( a). ² Every quadratic ¯eld extension corresponds to a subgroup of Gv of index 2, and hence to a homomorphism from Gv to Z=2Z. To prove the theorem, it must be shown that this is an inverse to the function recv, and that recv is a group homomorphism. We leave both of these facts as exercises for the reader. ¤ 1.2. Adeles and Ideles. The statement of global class ¯eld theory is signi¯cantly more di±cult than local class ¯eld theory. One side is expected: if G = Gal(Q¹ =Q), then we are interested in the group Hom(G; A), for any ¯nite abelian group A. But we do not ¯nd an isomorphism from Hom(G; A) to Hom(Q£;A). In this case, we must replace Q£ with the idele class group, which we explain here. Recall that for every \place" v (a prime number or 1), we have a ¯eld Qv (either £ Qp or R). Within the multiplicative group Qp , we have the compact open subgroup £ £ £ » Zp , and Qp =Zp = Z as abelian groups. We write Z1 = Q1 = R, for convenience. Q The ring of adeles (with an a, and denoted A), is de¯ned to be the subring of v Qv (the in¯nite product), whose elements (xv) satisfy: xv 2 Zv; for all but ¯nitely many indices v: One may check easily that A is a ring. It squeezes between the following two rings: Y Y Z^ £ R = Zv ½ A ½ Qv: v2V v2V LECTURE NOTES, WEEK 7 MATH 222A, ALGEBRAIC NUMBER THEORY 3 Note that Q is a subring of A. Indeed, if q 2 Q, then we may consider the sequence (qv), with qv = iv(q), where iv : Q ,! Qv is the unique inclusion of ¯elds, for all v. Note that (qv) is in A, since for every prime p not dividing the denominator of q, ip(qp) 2 Zp. Let i: Q ,! A denote the resulting embedding. Note that: i(Q) \ (Z^ £ R) = Z: The group of invertible elements of A is called the group of ideles (with an i), and denoted A£ (though some people use I). The group A£ can be seen as the Q £ subgroup of sequences (xv) 2 v Qv , satisfying: £ xv 2 Zv for all but ¯nitely many indices v: The ideles squeeze between the following two abelian groups: Y Y ^£ £ £ £ £ Z £ R = Zv ½ A ½ Qv : v2V v2V The ring embedding i: Q ,! A restricts to an embedding of abelian groups i: Q£ ,! A£. The quotient group A£=Q£ is called the idele class group. In order to understand it, we ¯rst prove the following: Proposition 1.4. The group of ideles satis¯es: A£ = (Z^£R£)i(Q£): £ £ Proof. Suppose that x = (xv) 2 A . Let S be the set of places at which xv 62 Zv . Then S Qis ¯nite, and 1 62 S, since x is an idele. Let np = val(xp), for all p 2 S. ¡np Let q = p2S p . Then we see that: val(xpip(q)) = 0; 81 6= p 2 V: Thus xi(q) 2 Z^£R£, and we are done. ¤ Corollary 1.5. There is an isomorphism: A£=i(Q£) =» (Z^£ £ R£)=i(Z£): 1.3. Global Class Field Theory. The main theorem of global class ¯eld theory may now be stated, using the idele class group. We abuse notation, and just view Q£ as a subgroup of A£, rather using the embedding i. Theorem 1.6. Let G = Gal(Q¹ =Q). Then, if A is any ¯nite abelian group, then there is a group isomorphism: rec(A) : Hom(G; A) ! Hom(A£=Q£;A): In the quadratic case, this follows from Theorems 3 and 4, of Chapter III, of Serre: Theorem 1.7. Let G = Gal(Q¹ =Q). Then there is a group isomorphism: (2) £ £ rec : Hom(G; Z=2Z) ! Homcont(A =Q ; Z=2Z): Proof. We begin by de¯ning the homomorphism rec(2). ² Suppose that γ 2 Hom(G; Z=2Z) is nontrivial. Equivalently, we may choose a subgroup G0 ½ G, of index 2. Equivalently, we may choose a quadratic extension K of Q. 4 MARTIN H. WEISSMAN ² Choose ap generator of K as a quadratic extension. Namely, we can write K = Q( a), for an element nonsquare element a 2 Q£, uniquely deter- mined up to Q£2. £ ² Suppose that x = (xv) 2 A . Then we may de¯ne: Y ´(x) = (a; xv)v: v This in¯nite product is well-de¯ned, since for all but ¯nitely many places, £ £ a 2 Zv and xv 2 Zv , and hence (a; xv)v = 1 (if v 6= 2; 1). This follows from Theorem 1, of Chapter III of Serre. ² If x 2 Q£, then we have ´(x) = 1, by the Product Formula (Theorem 3, of Chapter III of Serre). ² Hence ´, originally a homomorphism from A£ to Z=2Z, descends to a ho- momorphism´ ¹ from A£=Q£ to Z=2Z. ² ´ depends only on a modulo squares in Q£2, and hence only on the original choice of γ. ² Hence, associated to every γ 2 Hom(G; Z=2Z), we have a well-de¯ned element: rec(2)(γ) = ´ : A£=Q£ ! Z=2Z: Now, we show that rec(2) has an inverse: ² Suppose that we are given a nontrivial continuous homomorphism: ´ : A£=Q£ ! Z=2Z: ² Then, ´ is uniquely determined by its restriction to Z^£ £R£, and it is trivial on Z£ = f§1g. It follows that ´ is uniquely determined by its restriction Q ^£ £ ´^ to Z = p Zp . ² There exists a family of homomorphisms ´p, one for each prime number, such that: Y ´^(^x) = ´p(xp); p £ for everyx ^ = (xp) 2 Z^ . Moreover, there exists a ¯nite set S, such that ´p is nontrivial, if and only if p 2 S.