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Applications of Global Class Field Theory Course Notes

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Applications of Global Class Field Theory Course Notes Math 160c Spring 2013 Caltech Applications of Global Class Field Theory Course Notes Andrei Jorza Contents 1 Local Class Field Theory 2 1.1 Main results . 2 1.2 Application . 2 2 Glocal Class Field Theory 3 2.1 Adeles . 3 2.2 The Dirichlet unit theorem using adeles . 3 2.3 Main results . 4 2.4 Conductors . 4 2.5 Hilbert, ray and ring class elds . 5 3 Selmer groups and applications 7 3.1 Selmer systems and Selmer groups . 7 3.2 Global duality and dual Selmer groups . 8 3.3 Euler characteristics and sizes of Selmer groups . 9 4 Rational points on elliptic curves 12 4.1 Facts about elliptic curves . 12 4.2 Cohomology of elliptic curves over nite elds . 13 4.3 Descent for rational points . 14 4.4 Selmer groups for elliptic curves . 14 4.5 Mordell-Weil . 16 4.6 Standard proof of Mordell-Weil . 16 5 Characters with prescribed behavior 17 5.1 Grunwald-Wang . 17 5.2 Characters with prescribed nite order local behavior . 19 5.3 Characters with prescribed local behavior at innite places . 19 6 Projective Galois representations 22 6.1 A theorem of Tate . 22 6.2 Lifting projective Galois representations . 24 6.3 Local Galois representations in the tame case . 25 1 7 Iwasawa theory for -extensions 28 Zp 7.1 -extensions and Leopoldt's conjecture . 28 Zp 7.2 Class groups and Galois modules . 34 7.3 The Iwasawa algebra . 36 7.4 Modules over the Iwasawa algebra . 37 7.5 Class numbers in -extensions . 38 Zp 8 Hecke theory for GL(1) 40 8.1 Fourier analysis . 40 8.2 Local zeta integrals . 41 8.3 Local functional equation and local "-factors . 42 8.4 Global zeta integrals . 44 8.5 Global L-functions and "-factors . 45 8.6 Applications . 46 9 Hecke theory for Galois representations 48 9.1 Global theory . 48 9.2 Deligne's local "-factors . 50 Lecture 1 2013-04-01 1 Local Class Field Theory 1.1 Main results (1.1.1) Have an upper ramication ltration u for such that −1 , (−1;0] , (0;1] GK u ≥ −1 GK = GK GK = IK GK = . If is an algebraic extension then u u u . PK L=K GL=K = GK =(GK \ GL) (1.1.2) The invariant map is an isomorphism ∼ 2 × ∼ such that if then invK : Br(K) = H (GK ; K ) = Q=Z L=K invK ◦ cor = invL and invL ◦ res = [L : K] invK . (1.1.3) Tate duality. If is a nite -module let ∗ . The Galois group acts on M GK M = Hom(M; Q=Z) GK ∗ ∗ via ∗ ∗ −1 . Write ∼ . Write i i IK . m 2 M (gm )(m) = m (g m) M(1) = M ⊗Q=Z µ1 Hur(GK ;M) = H (GK =IK ;M ) Then there exists a perfect pairing i 2−i ∗ 2 ∗ 2 2 × H (GK ;M) ⊗ H (GK ;M (1)) ! H (GK ;M ⊗ M (1)) ! H (GK ; µ1) ! H (GK ; K ) ! Q=Z such that i ? 2−i ∗ . Hur(GK ;M) = Hur (GK ;M (1)) (1.1.4) The Artin map. For nite write −1 ×, 0 × and n n for . K=Qp UK = K UK = OK UK = 1 + ($) n ≥ 1 There exists a homomorphism × ∼ ab such that n n;ab. It has the property that rK : K = WK rK (UK ) = GK × _ × × rK (NL=K x) = rL(x) for x 2 L and rK (x) = cor (rL(x)) for x 2 K ⊂ L . 1.2 Application We will prove Kronecker-Weber for local elds. Theorem 1.1. The maximal abelian extension of is . Qp Qp(µ1) Zb Z ab ∼ ab Zb ab ∼ × Proof. Recall that GK = IK Frob and WK = IK Frob . Thus G = (I )Frob × Frob and I = o K o K Qp Qp p p Qp Zp and so Gab is a quotient of × which we will show to be equal to G under the reciprocity map. Qp Zp o Zb Qp(µ1)=Qp First, recall ur × and since '(n) if it follows that ur Qp = Qp(!(α)jα 2 Fp ) p ≡ 1 (mod n) p - n Qp = Qp(ζnjp - . Next is ramied over and so ur and therefore ur . This gives n) Qp(ζp) Qp ζp -Qp Qp \ Qp(µp1 ) = Qp G =∼ G × G ur . Qp(µ1)=Qp Qp(µp1 )=Qp Qp =Qp 2 2 Glocal Class Field Theory 2.1 Adeles (2.1.1) If is a nite extension then , ×;0 is the connected component of in × . For K=Q K1 = K ⊗Q R K1 1 K1 an embedding write and write . Then Q , × Q × v : K,! R v j R v : K,! C v j C K1 = vj1 Kv K1 = vj1 Kv and K×;0 = Q (0; 1) Q ×. 1 vjR vjC C (2.1.2) Write = Q0 K with the restricted product topology. For a nite set of places S write AK fOv g v K = Q K and S = Q0 K in which case = K × S . Then the ring = Q0 K has S v2S v AK v2 =S;fOv g v AK S AK AK fOv g v the product topology, is a discrete subgroup and is compact. K ⊂ AK AK =K × Q0 × × × S;× (2.1.3) Write = × K with the restricted product topology. As above = K × . The AK fOv g v AK S AK natural inclusion × is not continuous and in fact the topology on × is the subset topology induced AK ⊂ AK AK by the map × which takes to −1 . Write × . Dene 1 × as the AK ,! AK × AK x (x; x ) j · jAK : AK ! (0; 1) AK ⊂ AK kernel of . Then × 1 is a discrete subgroup, 1 is continuous and 1 × is compact. j · jAK K ⊂ AK AK ⊂ AK AK =K (2.1.4) Strong approximation states that if then S is dense. S 6= ; K ⊂ AK 2.2 The Dirichlet unit theorem using adeles Theorem 2.1. Let be a number eld with real embeddings and complex embeddings. Then × is K=Q r s OK a nitely generated abelian group of rank r + s − 1. Proof. For a nite set S of places which include the innite places write OK [1=S] = fx 2 Kjv(x) ≥ 0; v2 = Sg. Note that if then × × . We will show that × is a nitely generated abelian S = fv j 1g O[1=S] = OK OK [1=S] group of rank jSj − 1. Let ClS(K) be the class group of OK [1=S], i.e., the set of ideals modulo the set of principal ideals. An element of × gives the fractional ideal Q v(av ) of and the set of fractional ideals is a = (av) AK v2 =S($v) OK [1=S] isomorphic to × × Q ×. Write 1 Q in which case × × ∼ 1 1 . AK =KS v2 =S Ov KS = fx 2 KSj v2S jxvjv = 1g AK =KS = AK =KS It is easy to see that 1 × 1 Q × which gives the exact sequence ClS(K) = AK =K KS v2 =S Ov 1 Y × × 1 × 1 ! KS Ov =OK [1=S] ! AK =K ! ClS(K) ! 1 v2 =S Immediately one sees ClS(K) as a quotient of a compact group by an open subgroup and so ClS(K) is nite. Dene as the kernel of the summation map and write the kernel of . Then ∆ ⊕vR ! R ∆S ⊕v2SR ! R one has the map 1 × given by . Clearly Q × 1 is surjective log : AK =K ! ∆ (av) 7! (log javjv) v2 =S Ov KS ! ∆S and so get an exact sequence 1 Y × × × KS Ov =OK [1=S] ! ∆S= log OK [1=S] ! 0 v2 =S × which exhibits ∆S= log OK [1=S] as the image via a continuous map of an open subgroup of a compact group, × × therefore ∆S= log OK [1=S] is compact. In particular log OK [1=S] is a lattice in the (jSj − 1)-dimensional ∆S. × We would like to prove that OK [1=S] is a nitely generated abelian group of rank jSj − 1. To do this × it is enough to show that the intersection of the kernel of log with OK [1=S] consists only of torsion. What 1 1 Q × Q Q 1 is the kernel? The kernel of log on is f(av) 2 jjavjv = 1g, i.e., O × {±1g × S . AK AK v-1 v vjR vjC This is compact and its intersection with K× is compact and discrete, therefore nite. Since it is nite this intersection must be µ1(K), i.e., torsion. 3 2.3 Main results (2.3.1) The global Brauer sequence 0 ! Br(K) ! ⊕v Br(Kv) ! Q=Z ! 0 is exact where the rightmost map is the sum of the local invariant maps. Application: a global quaternion algebra over any number eld must be nonsplit at an even number of places. (Used to study Hilbert modular forms.) (2.3.2) The Artin map × ab has the property that where × rK : AK ! GK rK (1;:::; 1; x; 1;:::) = rKv (x) x 2 Kv is placed in position and is the local Artin map. It gives v rKv × × ×;0 ∼ ab rK : AK =K K1 = GK such that and induces an isomorphism ab ∼ × × ×. rL(x) = rK (NL=K (x)) rK GL=K = AK =K NL=K AL Lecture 2 2013-04-03 2.4 Conductors (2.4.1) If is a nite extension and is a continuous complex nite dimensional representation of K=Qp V GK Z 1 Gu dene the conductor cond(V ) = codimV (V K )du. Since the Galois action is continuous, V is xed by −1 an open subgroup of and so the integral is nite. For any , and if and GK V cond(V ) 2 Z≥0 cond(V ) = 0 only if V is unramied and cond(V ) ≤ 1 if and only if V is tamely ramied.
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