Chapter V

Local Class Field Theory

§ 1. The Local Reciprocity Law

The abstract class field theory that we have developed in the last chapter is now going to be applied to the case of a local field, i.e., to a field which is complete with respect to a discrete valuation, and which has a finite residue class field. By chap. II, (5.2), these are precisely the finite extensions K of the fields Qp or IF p«t». We will use the following notation. Let

VK be the discrete valuation normalized by vK(K*) = Z,

OK = {a E K I VK(a) 2: o} the valuation ring,

PK = {a E K I VK(a) > o} the maximal ideal,

K = OK /PK the residue class field,

UK = {a E K* I VK(a) = o} the unit group,

uin) = 1 + PK the group of n-th higher units, n = 1,2, ... ,

q = qK = #K,

lalp = q-vK(a) the normalized p-adic absolute value,

ILn the group of n-th roots of unity, and ILn(K) = ILn n K*.

7rK, or simply 7r, denotes a prime element of K, i.e., PK = 7r0K.

In local class field theory, the role of the profinite group G of abstract class field theory is taken by the absolute Galois group G(klk) of a fixed local field k, and that of the G-module A by the multiplicative group k* of the separable closure k of k. For a finite extension K Ik we thus have AK = K*, and the crucial point is to verify for the multiplicative group of a local field the class field axiom:

(1.1) Theorem. For a cyclic extension L IK of local fields, one has

#Hi(G(LIK),L*) ={[L:K] for~=O, 1 fon = -1.

J. Neukirch, Algebraic Number Theory© Springer-Verlag Berlin Heidelberg 1999

318 Chapter V. Local Class Field Theory

Proof: For i = -1 this is the claim of proposition (3.5) ("Hilbert 90") in chap. IV. So all we have to show is that the Herbrand quotient is h(G,L*) = #Ho(G,L*) = [L : K], where we have put G = G(LIK). The exact sequence

in which Z has to be viewed as the trivial G-module, yields, by chap. IV, (7.3),

h(G,L*) = h(G,Z)h(G,Ud = [L: KJh(G,Ud.

Hence we have to show that h (G , U L) = 1. For this we choose a normal basis {aO" I (Y E G} of LIK (see [93], chap. VIII, §12, th.20), a E OL, and consider in OL the open (and closed) G-module M = LO"EC oKaO". Then the open sets

v n = 1 +rr'KM, n = 1,2, ... ,

form a basis of open neighbourhoods of 1 in U L. Since M is open, we have rr~ OL ~ M for suitable N, and for n :::: N the V n are even subgroups (of finite index) of UL, because we have

(rrn M) (rrn M) = rr2n M M C rr2n 0 C rr 2n - N M C rrn M. KKK - K L_ K - K

Hence VnVn ~ V n, and since 1 - rr'KfJ." for fJ., E M, lies in V n, so

does (1 - rr'KfJ.,)-1 = 1 + rr'K(L~l/lJrr~(i-I)). Via the correspondence 1 + rr'Ka 1--+ a mod rrK M, we obtain G-isomorphisms as in II, (3.10),

V n jVn+1 ~ MjrrK M = EB (OK jpK )aO" = Indc(OK jpK). O"EC

So by chap. IV, (7.4), we have Hi(G, vnjvn+l) = 1 for i = 0, - 1 and n :::: N. This in turn implies that Hi (G, Vn) = 1 for i = 0, - 1 and n :::: N. Indeed, ifforinstance i = 0 and a E (vn)C, then a = (Ncbo)al, with bo E V n, al E (vn+I)C, and thus al = (Ncb l )a2, for some bl E V n+ l , a2 E (vn+2)C, etc.; in general,

ai = (NCbi)ai+l, bi E V n+i , ai+l E (Vn+i+I)C .

This yields a = Ncb, with the convergent product b = TI~o bi E V n, so that HO(G, vn) = 1. In the same way we have for a E V n such that Nca = 1, that a = bO"-I, for some b E V n, where (Y is a generator of G. Thus H-I(G, vn) = 1. We now obtain

because U L j V n is finite. o

§ 1. The Local Reciprocity Law 319

(1.2) Corollary. If L I K is an unramified extension of local fields, then for i = 0, - 1, one has

Hi(G(LIK),UL) =1 and Hi(G(LIK),Ut)) =1 for n=I,2, ...

In particular,

Proof: Let G = G(LIK). We have already seen that Hi(G,Ud = 1 in

chap. IV, (6.2). In order to prove Hi (G, ut)) = 1, we first show that

Hi(G,A*) = 1 and Hi(G,A) = 1,

for the residue class field A of L. It is enough to prove this for i = -1, as A is finite, and so h(G,A*) = h(G,A) = 1. We have H-I(G,A*) = 1 by Hilbert 90 (see chap. IV, (3.5)). Let f = [A : K] be the degree of A over the residue class field K of K, and let cp be the Frobenius automorphism of AI K. Then we have

f-I . f-I . #NGA = #{ x E A I L xrp' = L xq' = o} :::: qf- I

i=O i=O

and #(cp - 1)A = qf- I ,

since the map A rp-\ A has kernel K. Therefore H- I (G, A) = NG A/(cp - 1)A

=1. Applying now the exact hexagon of chap. IV, (7.1), to the exact sequence

of G-modules

1 --+ uil) --+ UL --+ A* --+ 1,

we obtain Hi (G, u2)) = Hi (G, U d = 1, because Hi (G, A *) = 1. If n is a prime element of K, then n is also a prime element of L, so the map

U t) -+ A given by 1 + ann H- a mod P L is a G -homomorphism. From the exact sequence

1 --+ U(n+1) --+ U(n) --+ A --+ 1 L L '

we now deduce by induction just as above, because Hi (G, A) = 0, that

Hi(G,Uin+l)) = Hi(G,Ut)) = 1,

since Hi (G, U2)) = 1. o

320 Chapter V. Local Class Field Theory

We now consider the maximal unramified extension klk over the ground field k. By chap. II, §9, the residue class field of k is the algebraic closure iZ of the residue class field K of k. By chap. II, C9.9), we get a canonical isomorphism

GCklk) ~ GCiZIK) ~ Z. It associates to the element 1 E Z the Frobenius automorphism x t--+ xq

in GCiZIK), and the Frobenius automorphism qJk in GCklk) which is given by

arpk == aq mod Pk' a E ok'

For the absolute Galois group G = GCklk) we therefore obtain the continu­ous and surjective homomorphism

d:G----+Z.

Thus the abstract notions of chap. IV, § 4, based on this homomorphism, like "unramified", "ramification index", "inertia degree", etc., do agree, in the case at hand, with the corresponding concrete notions defined in chap. II.

As stated above we choose A = k* to be our G-module. Hence AK = K*, for every finite extension K I k. The usual normalized exponential valuation Vk : k* -+ Z is then henselian with respect to d, in the sense of chap. IV, C 4.6). For, given any finite extension K Ik, -L VK is the extension of Vk to K*, and

eK by chap. II, (4.8),

1 * 1 * 1 * -vKCK ) = [K k] vk(NKlk K ) = -j-Vk(NK1kK ), eK : eK K

i.e., vkCNKlkK*) = /KVKCK*) = /KZ. The pair of homomorphisms

(d : G -+ Z, Vk : k* -+ Z)

therefore satisfies all the properties of a class field theory, and we obtain the Local Reciprocity Law:

(1.3) Theorem. For every finite Galois extension L I K of local fields we have a canonical isomorphism

rLIK : GCLIK)ab ~ K*/NLIKL*.

The general definition of the reciprocity map in chap. IV, (5.6), was actually inspired by the case of local class field theory. This is why it is especially transparent in this case: let a E G~L IK), and let (j be an extension of a to the maximal unramified extension L I K of L such that dK C (j) E N

§ 1. The Local Reciprocity Law 321

ior, in other words, a Ii = fPK' for some n EN. If E is the fixed field of a and rr E E E is a prime element, then

rLIK(cr) = NEIK(rrI;) mod NLIKL*.

Inverting rLIK gives us the local norm residue symbol

( ,L IK) : K* ---+ G(L IK)ab.

It is surjective and has kernel N L IK L * .

In global class field theory we will have to take into account the field ~ = Qoo along with the p-adic number fields Qp. It also admits a reciprocity law: for the unique non-trivial Galois extension C I~, we define the norm residue symbol

( ,q~): ~* ---+ G(q~)

by (a, C I~)R = Rsgn(a) .

The kernel of ( ,C I~) is the group ~~ of all positive real numbers, which is again the group of norms Nq]RC* = {zz I Z E C*}.

The reciprocity law gives us a very simple classification of the abelian extensions of a local field K. It is formulated in the following

(1.4) Theorem. The rule

L ~ NL = NLIKL*

gives a I-I-correspondence between the finite abelian extensions of a local field K and the open subgroups N of finite index in K * . Furthermore,

Ll ~ L2 {::::=:} NLI ;2 N L2 , NLIL2 = NLI nNL2' NLlnL2 = NL\NL2 ·

Proof: By chap. IV, (6.7), all we have to show is that the subgroups N of K* which are open in the norm topology are precisely the subgroups of finite index which are open in the valuation topology. A subgroup N which is open in the norm topology contains by definition a group of norms N L IK L *. By (1.3), this has finite index in K*. It is also open because it contains the subgroup NLIKUL which itself is open, for it is closed, being the image of the compact group U L, and has finite index in UK. We prove the converse first in

The case char(K) f n. Let N be a subgroup of finite index n = (K* : N). Then K*n ~ N, and it is enough to show that K*n contains a group of

322 Chapter V. Local Class Field Theory

norms. For this we use Kummer theory (see chap. IV, §3). We may assume that K* contains the group ILn of n-th roots of unity. For if it does not, we put K J = K (ILn). If K;n contains a group of norms NLJlKI q, and L IK is a Galois extension containing L J, then

NLIKL* = NKJlK (NLIKI L*) ~ NKJlK (NL] IK]L"i)

~ NKJlK(K;n) ~ K*n.

So let ILn ~ K, and let L = K (!;fi{*) be the maximal abelian extension of exponent n. Then by chap. IV, §3, we have

Hom(G(L IK), ILn) ~ K* / K*n .

By chap. II, (5.8), K* / K*n is finite, and then so is G(L IK). Since K* /NLIK L * is isomorphic to G(L IK) and has exponent n, we have that K*n £;; N LIK L *, and (*) yields

and therefore K*n = NLIKL*.

The case char(K) = pin. In this case the proof will follow from Lubin-Tate theory which we will develop in § 4. But it is also possible to do without this theory, at the expense of ad hoc arguments which turn out to be somewhat elaborate. Since the result has no further use in the remainder of this book, we simply refer the reader to the beautiful treatment in [122], chap. XI, §5, and chap. XIV, § 6. D

The proof also shows the following

(1.5) Proposition. If K contains the n -th roots of unity, and if the character­istic of K does not divide n, then the extension L = K ( !;fi{*) I K is finite, and one has

Theorem (1.4) is called the existence theorem, because its essential statement is that, for every open subgroup N of finite index in K *, there exists an abelian extension L I K such that N L IK L * = N. This is the "class field" of N. (Incidentally, when char(K) = 0, every subgroup of finite index is automatically open - see chap. II, (5.7).) Every open subgroup of K* contains some higher unit group uj;), as these form a basis of

neighbourhoods of 1 in K*. We put U<t) = UK and define:

§ 1. The Local Reciprocity Law 323

(1.6) Definition. Let L I K be a flnite abelian extension, and n the smallest

number:::: 0 such that U j;) S; N L I K L *. Then the ideal

f= PK is called the conductor of L I K .

(1.7) Proposition. A flnite abelian extension L IK is unramifled if and only if its conductor is f = 1.

Proof: If LIK is unramified, then UK = NLIKUL by (1.2), so that f = 1. If conversely f = 1, then UK S; NLIKUL and nf( E NLIKL*, for n = (K* : NLIKL*). If MIK is the unramified extension of degree n, then NMIKM* (nf() x UK S; NLIKL*, and then M :2 L, i.e., LIK is unramified. D

Every open subgroup N of finite index in K* contains a group of the form (n!) x uj;). This is again open and of finite index. Hence every finite abelian extension L I K is contained in the class field of such a group (n!) x uj;). Therefore the class fields for the groups (n!) x uj;) are particularly important. We will characterize them explicitly in §5, as immediate analogues of the cyclotomic fields over Qp. In the case of the

ground field K = Qp' the class field of the group (p) x uj;) is precisely the field Qp(Mpn) of pn-th roots of unity:

(1.8) Proposition. The group of norms of the extension Q p (M pn ) I Q p is the

group (p) x U~;.

Proof: Let K = Qp and L = Qp(Mpn). By chap. II, (7.13), the extension L I K is totally ramified of degree pn-l (p - 1), and if s is a primitive pn -th root of unity, then 1 - s is a prime element of L of norm N L I K (l - n = p. We now consider the exponential map of Qp. By chap. II, (5.5), it gives an isomorphism

exp : P~ ----+ ui;) for v :::: 1, provided p =f. 2, and for v :::: 2, even if p = 2. It transforms the isomorphism P~ -+ p~+S-1 given by a J--+ pS-l(p - l)a,

into the isomorphism ui;) -+ ui;+s-l) given by x J--+ x PS - 1(p-l), so that

(U (1))pn-l(p_I) - U(n)'f -'- 2 d (U(2))2n- 2 - U(n)'f - 2 1 K - K 1 P't'" ,an K - K 1 p- ,n>

324 Chapter V. Local Class Field Theory

(the case p = 2, n = 1 is trivial). Consequently, we have uj;) ~ NLIKL* if p =I=- 2. For p = 2 we note that

uf) = uf) U 5uf) = (uf)) 2 U 5{ uf)) 2,

because a number that is congruent to 1 mod 4 is congruent to 1 or 5 mod 8. Hence

(n) (2)) 2n- 1 2n- 2 ( (2)) 2n- 1 UK = UK U5 UK .

It is easy to show that 52n - 2 = NLIK(2+i), so uj;) ~ NLIKL* holds also in

case p = 2. Since p = NLIK(l - n, we have (p) x uj;) ~ NLIKL*, and since both groups have index pn-l (p - 1) in K*, we do find that

NLIKL* = (p) x uj;) as claimed. 0

As an immediate consequence of this last proposition, we obtain a local version of the famous theorem of Kronecker-Weber, to the effect that every finite abelian extension of Q is contained in a cyclotomic field.

(1.9) Corollary. Every finite abelian extension of L I Q p is contained in a field Qp(n, where ~ is a root of unity. In other words:

The maximal abelian extension Q~b I Q p is generated by adjoining all roots of unity.

Proof: For suitable f and n, we have (pi) x u~n; ~ NLIKL*. Therefore L

is contained in the class field M of the group

(pi) x U~; = (pi) x UQp) n (p) x U~;).

By (1.4), M is the composite of the class field for (pi) x UQp - this being

the unramified extension of degree f - and the class field for (p) x U~;. M is therefore generated by the (pi - 1) pn -th roots of unity. 0

From the local Kronecker-Weber theorem, one may readily deduce the global, classical Theorem of Kronecker-Weber.

(1.10) Theorem. Every finite abelian extension L IQ is contained in a field Q(n generated by a root of unity ~.

§ 1. The Local Reciprocity Law 325

Proof: Let S be the set of all prime numbers p that are ramified in L, and let Lp be the completion of L with respect to some prime lying above p. Then LplQp is abelian, and therefore Lp ~ Qp(f.1,np)' for a suitable np. Let pep be the precise power of p dividing np , and let

n = n pep. PES

We will show that L ~ Q(JLn). For this let M = L(JLn). Then MIQ is abelian, and if p is ramified in M I Q, then p must lie in S. If M p is the completion with respect to a prime of M above p whose restriction to L gives the completion L p, then

Mp = Lp(f.1,n) = Qp(f.1,pep n') = Qp(JLpep) Qp(f.1,n')'

with (n/,p) = 1. Qp(JLn')IQp is the maximal unramified subextension of Q p (JL pep n' ) I Q p' The inertia group [p of M pi Q p is therefore isomorphic to the group G(Qp(JLpep )IQp)' and consequently has order q;(pep), where q; is Euler's function. Let [ be the subgroup of G(MIQ) generated by all [p,

pES. The fixed field of [ is then unramified, and hence by Minkowski's theorem from chap. III, (2.18), it equals Q, i.e., [ = G(MIQ). On the other hand we have

#[ :::: n #[p = n q;(pep) = q;(n) = [Q(f.1,n) : Q], PES PES

and therefore [M : Q] = [Q(JLn) : Q], so that M = Q(JLn). This shows that L ~ Q(f.1,n). D

The following exercises 1-3 presuppose exercises 4-8 of chap. IV, § 3.

Exercise 1. For the Galois group r = GCKIK), one has canonically

HIcr,'ll/n'll) ~ 'll/n'll and HIcr,JLn) ~ UKK*n/K*n,

the latter provided that n is not divisible by the residue characteristic.

Exercise 2. For an arbitrary field K and a G K -module A, put

HICK,A) = HI(GK,A).

If K is a p -adic number field and n a natural number, then there exists a nondegen­erate pairing

of finite groups given by (x,a) 1-+ x((a, KIK».

If n is not divisible by the residue characteristic p, then the orthogonal complement of

H~rCK,'ll/n):= HI(G(KIK),'ll/n'll) ~ HI(K,'ll/n'll)

is the group

326 Chapter V. Local Class Field Theory

Exercise 3. If LIK is a finite extension of p-adic number fields, then one has a commutative diagram

HI(L,ZlnZ) x HI(L, /Ln) ~ Zln'!!,

i lNL'K II HI(K, ZlnZ) x HI(K, /Ln) ~ ZlnZ.

Exercise 4 (Local Tate Duality). Show that the statements of exercises 2 and 3 generalize to an arbitrary finite G K -module A instead of Z I nZ, and A' = Hom(A, K*) instead of /Ln.

Hint: Use exercises 4-8 of chap. IV, §3.

Exercise 5. Let L I K be the composite of all Z p -extensions of a p -adic number field K (i.e., extensions with Galois group isomorphic to Zp). Show that the Galois group G(LIK) is a free, finitely generated Zp-module and determine its rank.

Hint: Use chap. II, (5.7).

Exercise 6. There is only one unramified Zp-extension of K. Generate it by roots of unity.

Exercise 7. Let p be the residue characteristic of K, and let L be the field generated by all roots of unity of p-power order. The fixed field of the torsion subgroup of G (L I K) is a Z p -extension. It is called the cyclotomic Z p -extension.

Exercise 8. Let ijplQp be the cyclotomic Zp-extension ofQp, let G(ijpIQp) ;:: Zp

be a chosen isomorphism, and let a: GQp ~ Zp be the induced homomorphism of the absolute Galois group. Show:

For a suitable topological generator u of the group of principal units of Qp,

A log a v(a) = --,

logu

defines a henselian valuation with respect to a, in the sense of abstract p -class field theory (see chap. IV, §5, exercise 2).

Exercise 9. Determine all p-class field theories (d : GK ~ Zp, v : K* ~ Zp) over a p-adic number field K.

Exercise 10. Determine all class field theories (d : GK ~ Z, v : K* ~ Z) over a p-adic number field K.

Exercise 11. The Weil group of a local field K is the preimage W K of Z under the mapping dK : GK ~ Z. Show:

The norm residue symbol ( , K ab I K) of the maximal abelian extension K ab I K yields an isomorphism

which maps the unit group UK onto the inertia group I (K ab IK), and the group of principal units ui/) onto the ramification group R(K ab IK).

§ 2. The Norm Residue Symbol over Qp 327

§ 2. The Norm Residue Symbol over Qp

If ~ is a primitive m-th root of unity, with Cm, p) = 1, then QpCnlQp is unramified, and the norm residue symbol is obviously given by

But if ~ is a primitive pn -th root of unity, then we obtain the norm residue symbol for the extension Qp(nlQp explicitly in the simple form

(a,Qp(nIQp)~ = ~u-l,

where a = upvp(a) , and ~u-l is the power ~r with any rational integer r == u- l mod pn. This result is important, not only in the local situation, but it will play an essential role when we develop global class field theory (see chap. VI, § 5). Unfortunately, there is no direct algebraic proof of this fact known to date. We have to invoke a transcendental method which makes use of the completion K of the maximal unramified extension K of a local field K. We extend the Frobenius cp E G(K IK) to K by continuity. First we prove the

(2.1) Lemma. For every c E of(, resp. every c E U f(, the equation

xl{! - x = c, resp. Xl{!-l = C,

admits a solution in of(, resp. in U f(. If xl{! = x for x E of(, then x E OK.

Proof: Let rr: be a prime element of K. Then rr: is also a prime element of K, and we have the cp-invariant isomorphisms

U~/U(1) ~ -* K f( - K ,

(see chap. ~ (3.10)). Let C E Uf( and c = c mod Pf(' Since the residue class field iC of K is algebraically closed, the equation xl{! = x q = x . c (q = qK) has a solution =f. 0 in iC = of( /Pf(, i.e.,

I{!-l U c = Xl al , Xl E f( ,

For similar reasons, we find that al = Xf-l a2, for some X2 E ujp and

a2 E U'j), so that c = (Xlx2)1{!- l a2. Indeed, putting al = 1 + bIrr:,

1 . I-I{! 1 (I{! b ) d 2 . X2 = + Y2rr:, gIves alX2 == - Y2 - Y2 - I rr: mo rr:, I.e., we have to solve the congruence yf - Y2 - b l == 0 mod rr:, or equivalently the

328 Chapter V. Local Class Field Theory

equation yi - Y2 - bl = 0 in K. This is possible because K is algebraically closed. Continuing in this way, we get

x E U~n-l) n K '

U(n) an E k '

and passing to the limit finally gives c = x<fJ- l , where x = n;:o= I Xn E Uk.

The solvability of the equation x<fJ - x = c follows analogously, using the isomorphisms pi/pi+l ~ K.

Now let x E ok and x<fJ = x. Then, for every n 2: 1, one has

Indeed, for n = 1 we have x = a + nb, with a E oK' b E ok, and x<fJ = x

implies a<fJ == a mod n. Hence a = XI +nc, with XI E OK, c E OK' and therefore x = xI+n(b+c) = xI+nYl, YI E ok. The equation x = xn+nnYn

implies furthermore that y~ = Yn, so that we get as above Yn = Cn + ndn,

with Cn E OK, dn E Ok, and therefore x = (xn + cnnn) + nn+ldn = n+l -I' N . th 1· . xn+l+n Yn+l,lorsomexn+IEoK,Yn+lEOk. owpassmgto e 1m1t

in the equation (*) gives x = limn-+oo Xn E OK, because K is complete. 0

For a power series F(X 1, ••• ,Xn) E 0k[[Xl , ... ,Xn]], let F<fJ be the power series in Ok [[XI, ... ,XnJJ which arises from F by applying <p to the coefficients of F. A Lubin-Tate series for a prime element n of K is by definition a power series e(X) E OK[[X]] with the properties

e(X) == n X mod deg 2 and e(X) == xq mod n,

where q = qK denotes, as always, the number of elements in the residue class field of K. The totality of all Lubin-Tate series is denoted by £17:. In £17: there are in particular the polynomials

e(X) = uxq + n(aq_lXq-1 + ... + a2X2) + n X,

where u, ai E OK and u == 1 mod n. These are called the Lubin-Tate polynomials. The simplest one among them is the polynomial xq + n X. In the case K = CQ p for example, e(X) = (1 + X)P - 1 is a Lubin-Tate polynomial for the prime element p.

(2.2) Proposition. Let n and Tf be prime elements of K, and let e(X) E £17:, e(X) E £-if be Lubin-Tate series. Let L(X I, ... , Xn) = L?=I aiXj be a linear form with coefficients aj E Ok such that

§ 2. The Nonn Residue Symbol over {Qlp 329

Then there is a uniquely detennined power series F (X I, ... , Xn) E 0k[[XI , ... , Xn]] satisfying

F(X I , ... ,Xn)::=L(XI , ... ,Xn )moddeg2,

e(F(XI , ... ,Xn)) = FCI'(e(X I ), ... ,e(Xn)).

If the coefficients of e, e, L lie in a complete subring 0 of ok such that oCl' = 0,

then F has coefficients in 0 as well.

Proof: Let 0 be a complete subring of ok such that oCl' = 0, which contains the coefficients of e, e, L. We put X = (XI, ... , Xn) and e(X) = (e(X I ), ... , e(Xn)). Let

00

F(X) = L Ev(X) E o[[X]] v=1

be a power series, Ev(X) its homogeneous part of degree v, and let

r

Fr(X) = L Ev(X). v=1

Clearly, F (X) is a solution of the above problem if and only if FI (X) = L(X) and

(1) e(Fr(X)) ::=F;(e(X)) moddeg(r+l)

for every r > 1. We detennine the polynomials Ev(X) inductively. For v = 1 we are forced to take E 1(X) = L(X). Condition (1) is then satisfied for r = 1 by hypothesis. Assume that the Ev(X), for v = 1, ... , r, have already been found, and that they are uniquely determined by condition (1). We then put Fr+I(X) = Fr(X) + Er+I(X) with a homogeneous polynomial Er+1 (X) E o[X] of degree r + 1 which has yet to be determined. The congruences

e( Fr+1 (X)) ::= e( Fr (X)) + 7r Er+1 (X) mod deg(r + 2),

F:+ I (e(X)) ::= F; (e(X)) + jfr+1 E~+I (X) mod deg(r + 2)

show that Er+1 (X) has to satisfy the congruence

(2) G r+1 (X) + 7r Er+1 (X) - jfr+1 E~+I (X) ::= 0 mod deg(r + 2)

with Gr+I(X) = e(Fr(X)) - F;(e(X)) E o[[X]]. We have Gr+I(X) ::= 0 mod deg(r + 1) and

(3)

330 Chapter V. Local Class Field Theory

because e(X) == e(X) == xq mod Jr and arfJ == a q mod Jr for OlEO.

Now let Xi = X~l ... x~n be a monomial of degree r + 1 in o[X]. By (3), the coefficient of Xi in G r +! is of the form -JrfJ, with fJ E o. Let a be the coefficient of the same monomial Xi in Er+!. Then Jra - JrarfJ is the coefficient of Xi in Jr Er+1 - 7f E;+I' Since Gr+1 (X) == 0 mod deg(r + 1), (2) holds if and only if the coefficient a of Xi in Er+! satisfies the equation

(4) for every monomial Xi of degree r + 1. This equation has a unique solution a in 0i, which actually belongs to o. For if we put y = Jr- I 7fr +!, we obtain the equation

a - yarfJ = fJ ,

which is clearly solved by the series

a = fJ + yfJrfJ + y!+rfJfJrfJ'l>. + ... EO

(the series converges because vi(y) :::: O. If a' is another solution, then a - a' = y(arfJ - a'rfJ), hence vi(a - a') = vi(y) + vi«a -a')rfJ) = vi(y) + Vi (a - a'), i.e., vi(a - a') = 00 because vi(y) :::: 1, and therefore a = a'. As a consequence, for every monomial Xi of degree r + 1, equation (4) has a unique solution a in 0, i.e., there exists a unique Er+! (X) E o[X] satisfying (2). This finishes the proof.

o

(2.3) Corollary. Let Jr and jf be prime elements of K, and let e E En, e E Elf be Lubin-Tate series with coefficients in OK. Let Jr = U 7f, U E UK, and U = c: rfJ -!, c: E Ui. Then there is a uniquely determined power series e(X) E 0i[[X]] such thate(X) == c:X mod deg2 and

eo e = erfJ 0 e.

Furthermore, there is a uniquely determined power series [u](X) E OK [[X]] such that [u](X) == uX mod deg 2 and

eo [u] = [u] 0 e.

They satisfy e rfJ = e 0 [u].

Proof: Putting L(X) = c:X, we have JrL(X) = 7fLrfJ(X) and the first claim follows immediately from (2.2). In the same way, with the linear form L(X) = uX, one obtains the existence and uniqueness of the power series [u](X) E 0K[[X]]. Finally, defining e! = erfJ - 1

0 [u], we get

eo e! = (e 0 e)rfJ-1 0 [u] = (erfJ 0 e)rfJ- 1 0 [u] = (erfJ - 1 0 [u])rfJ 0 e = ei 0 e,

and thus e! = e because of uniqueness. Hence erfJ = e 0 [u]. 0

§ 2. The Norm Residue Symbol over (llp 331

(2.4) Theorem. Let a = upvp(a) E (ll;, and let t; be a primitive pn -th root of unity. Then one has

Proof: As N is dense in Zp, we may assume that u E N, (u, p) = 1. Let K = Qp' L = Qp(t;), and let 0' E G(LIK) be the automorphism defined by

-I t;a = t;U .

Since Qp(t;)IQp is totally ramified, we have G(LIK) = G(LIK), and

we view 0' as an element of G(iIK). Then 0- = O'<jJL E Frob(LIK) is an element such that dK(o-) = 1 and o-IL = 0'. The fixed field E of 0- is totally ramified because JEIK = dK(o-) = 1 by chap. IV, (4.5). The proof of the theorem is based on the fact that the field E can be explicitly generated by a prime element Jr E which is given by the power series e of (2.3).

In order to do this, assume 0- and <jJ = <jJL have been extended continuously to the completion L of L, and consider the two Lubin-Tate polynomials

e(X) = upX + xP and e(X) = (l + X)P - 1

as well as the polynomial [u](X) = (1 + X)U - 1. Then e([u](X» (1 + X)UP - 1 = [u ](e (X». By (2.3), there is a power series e(X) E ok [[X]] such that

eoe=e'P oe and e'P=eo[u].

Substituting the prime element "A = t; - 1 of L, we obtain a prime element of E by

7rx; = e("A).

Indeed, [u]("Aa) = (l + "Aa)u - 1 = t;au - 1 = t; - 1 = "A, and therefore

Jrf = e'P("Aa) = e([u]("Aa») = e("A) = JrE,

i.e., Jr E E E. We will show that

P(X) = en-\X)p-l + up E Zp[X]

is the minimal polynomial of Jr E, where ei (X) is defined by eO (X) = X and ei(X) = e(ei-1(X». P(X) is monic of degree pn-l(p -1) and irreducible by Eisenstein's criterion, as e(X) == XP mod p, and so en-1(X)p-l _ Xpn-1Cp_1) mod p. Finally, en(X) = en-1(X) . (up + en-1(X)p-l) en-1(X)P(X), so that

P(JrE)en- 1 (JrE) = en (JrE) . . . i' iii i

Since el(JrE) = el(e("A» = e'P (el("A» = e'P «(1 +"A)P -1) = e'P (t;P -1), we have en(JrE) = 0, en-I (JrE) i= 0, and thus P(JrE) = 0.

332 Chapter V. Local Class Field Theory

Observing that N LIK (~-1) = (_I)d p, d = [L : K] (see chap. II, (7.13», we obtain

NEIK(1fE) = (_I)dp(O) = (-I)dpu == U mod NLIKL*

and therefore rLIK(O') = u mod NLIKL*, i.e., (u,LIK) = (a,LIK) = 0', as required. 0

In order to really understand this proof of theorem (2.4), one has to read § 4. Let us note that one would get a direct, purely algebraic proof, if one could show without using the power series () that the splitting field of the polynomial en (X) is abelian, and that its elements are all fixed under (j = O'qJL. This splitting field would then have to be equal to the field E and every zero of P(X) = en (X)/en- 1(X) would have to be a prime element 1fE E E such that NEIK(1fE) == U mod NLIKL*, in which case rLIK(O') == u mod NLIKL*, and so (u,LIK) = 0'.

Exercise 1. The p-class field theory (d : GQp -+ 7l..p, v : Q; -+ 7l..) for the unramified

7l..p -extension of Qp' and the p-class field theory (d : GQp -+ 7l..p, v : Q; -+ 7l..p) for the cyclotomic 7l..p -extension of Qp (see § 1, exercise 7) yield the same norm residue symbol ( ,LIK).

Hint: Show that this statement is equivalent to formula (2.4): (u, Q/O IQpH" = ~u-l.

Exercise 2. Let LIK be a totally ramified Galois extel1§ion, andJet L (resp. K) be the completion <i the J,!laximal unramified ex~nsion L (resp. K) of L (resp. K). Show that NtIKL* = K*, and that every Y E L* with NtIK(Y) = 1 is of the form

Y = n zr;-I, ai E G(LIK).

Exercise 3 (Theorem of DWORK). Let L I K be a totally ramified abelian extension of p-adic number fields. Let x E K* and Y E L* such that NtIK(Y) = x. Let Zi E L* and choose ai E G(LIK) such that

'PK-I - TI 0";-1 Y - zi . i

Putting a = n ai, one has (x, LIK) = a-I.

Hint: See chap. IV, §5, exercise 1.

Exercise 4. Deduce from exercises 2 and 3 the formula (u,Qp(~)IQp)~ = ~u-l, for some pn -th root of unity ~.

§ 3. The Hilbert Symbol 333

§ 3. The Hilbert Symbol

Let K be a local field, or K = JR, K = C. We assume that K contains the group JLn of n-th roots of unity, where n is a natural number which is relatively prime to the characteristic of K (i.e., n can be arbitrary if char(K) = 0). Over such a field K we then have at our disposal, on the one hand, Kummer theory (see chap. IV, §3), and on the other, class field theory. It is the interplay between both theories, which gives rise to the "Hilbert symbol". This is a highly remarkable phenomenon which will lead us to a generalization of the classical reciprocity law of Gauss, to n-th power residues.

Let L = K ( ~) be the maximal abelian extension of exponent n. By (1.5), we then have

NLIKL* = K*n,

and class field theory gives us the canonical isomorphism G(LIK) ~ K*jKM.

On the other hand, Kummer theory gives the canonical isomorphism Hom(G(L IK), JLn) ~ K* j K*n .

The bilinear map

G(LIK) x Hom(G(LIK),JLn) -+ JLn, (a,X) 1------+ x(a),

therefore defines a nondegenerate bilinear pairing

(t) : K*jK*n x K*jK*n -+ JLn

(bilinear in the multiplicative sense). This pairing is called the Hilbert symbol. Its relation to the norm residue symbol is described explicitly in the following proposition.

(3.1) Proposition.

by

For a, b E K*, the Hilbert symbol (a;,b)

(a,K(~)IK) ~ = (a~b)~.

E JLn is given

Proof: The image of a under the isomorphism K* j K*n ~ G(L IK) of class field theory is the norm residue symbol a = (a, L IK). The image of b under the isomorphism K* j KM ~ Hom(G(L IK), JLn) of Kummer theory is the character Xb : G(LIK) ~ JLn given by Xb(i) = i~j~. By definition of the Hilbert symbol, we have

(a/) = Xb(a) = a~ j~, hence (a,K(~)IK)~ = (a,LIK)~ = (a;,b)~. 0

334 Chapter V. Local Class Field Theory

The Hilbert symbol has the following fundamental properties:

(3.2) Proposition.

(i) (aa;,b) = (apb)(a~b),

(ii) (a,;b') = (apb)(aj,b'),

(iii) (apb) = 1 {::=:} a is a nonn from the extension K ( ';fb) I K ,

(iv) (apb) = (bpa)-l,

(v) (a, \,-a) = 1 and (a, ;a) = 1,

(vi) If (apb) = 1 for all b E K*, then a E K*n.

Proof: (i) and (ii) are clear from the definition, (iii) follows from (3.1), and (vi) refonnulates the nondegenerateness of the Hilbert symbol.

If b E K* and x E K such that xn - b =1= 0, then

n-l

xn - b = TI (x - ~i 13), f3n = b, i=O

for some primitive n -th root of unity ~ . Let d be the greatest divisor of n such that yd = b has a solution in K, and let n = dm. Then the extension K (13) IK is cyclic of degree m, and the conjugates of x - ~i 13 are the elements x - ~ j 13 such that j == i mod d. We may therefore write

d-l

xn - b = TI NK({J)IK(X - ~i 13). i=O

Hence xn - b is a nonn from K ( ';fb) I K, i.e.,

( xn - b,b) _ -1.

P Choosing x = 1, b = 1 - a, and x = 0, b = -a then yield (v). (iv) finally follows from

§ 3. The Hilbert Symbol 335

In the case K = lit we have n = 1 or n = 2. For n = lone finds, of course, (a;,b) = 1, and for n = 2 we have

( a,b) sgna-l. sgnb-I - = (-1) 2 2,

P sgna-J

because (a, lit ( Jb) lilt) = 1 for b > 0, and = (-1) -2 - for b < O. Here the letter p symbolically stands for an infinite place.

Next we determine the Hilbert symbol explicitly in the case where K is a local field (# lit, C) whose residue characteristic p does not divide n. We call this the case of the tame Hilbert symbol. Since /.Ln S; /.Lq-I one has n I q - 1 in that case. First we establish the

(3.3) Lemma. Let (n,p) = 1 and x E K*. The extension K($)IK is unramified if and only if x E UK KM.

Proof: Let x = uyn with u E UK, Y E K*, so that K($) = K(:.yu). Let K' be the splitting field of the polynomial xn - u mod p over the residue class field K, and let K 'I K be the unramified extension with residue class field K' (see chap. II, §9, p. 173). By Hensel's lemma, Xn - u splits over K' into linear factors, so K ( :.yu) S; K I is unramified. Assume conversely that L = K ($) is unramified over K, and let x = urrr , where u E UK and rr is a prime element of K. Then vLC:::';urr r ) = *vLCrr r ) = ~ E Z, hence nlr, i.e., rrr E K*n, and thus x E UKKM. 0

Since UK = /.Lq-I x Uk!), every unit u E UK has a unique decomposition

u = w(u)(u)

with w(u) E /.Lq-I and (u) E uf), u == w(u) mod p. With this notation we will now prove the

(3.4) Proposition. If(n,p) = 1 anda,b E K*, then

(a~ b) = w( (_I)a~ ~; ) (q-I)/n ,

where a = vK(a), fJ = vKCb).

336 Chapter V. Local Class Field Theory

Proof: The function

( ba ) (q-l)/n (a, b) := w (_I)a.8 a.8

is obviously bilinear (in the multiplicative sense). We may therefore assume that a and b are prime elements: a = rr, b = -rru, u E UK. Since clearly

(rr, -rr) = Cr, ;1l") = 1, we may restrict to the case a = rr, b = u. Let

y = !;jU and K' = K(y). Then we have

(rr,u) =w(u)(q-l)/n and (rr,K'IK)y= (rr~u)y.

By (3.3), we see that K'IK is unramified and by chap.N, (5.7), (rr,K'IK) is the Frobenius automorphism ({J = ({J K' IK. Consequently,

(rr ~ U) = ({J: == yq-l == u(q-l)/n == w(u)(q-l)/n == (rr, u) mod p,

hence (1l"pU) = (rr,u), because JLq-l is mapped isomorphically onto K* by

UK ---+ K*. D

The proposition shows in particular that the Hilbert symbol

(rr ~ U) = w(u)(q-l)/n

(in the case (n, p) = 1) is independent of the choice of the prime element rr. We may therefore put

(~) := (rr~u) for u E UK.

( ~) is the root of unity determined by

(~) == u(q-l)/n mod PK.

We call it the Legendre symbol, or the n-th power residue symbol. Both names are justified by the

(3.5) Proposition. Let (n, p) = 1 and u E UK. Then one has

(~) = 1 {:::=} u is an n-th power mod PK.

§ 3. The Hilbert Symbol 337

Proof: Let ~ be a primitive (q - 1)-th root of unity, and let m = q ~ I. Then ~n is a primitive m-th root of unity, and

(~) = w(u)m = 1 {=::} w(u) E ILm {=::} w(u) = (~n)i

{=::} u == w(u) = (~i)n mod PK. 0

It is an important, but in general difficult, problem to find explicit formulae

for the Hilbert symbol (a;,b) also in the case pin. Let us look at the case

where n = 2 and K = Qp. If a E Z2, then (_l)a means

(_l)a = (-It,

where r is a rational integer == a mod 2.

(3.6) Theorem. Let n = 2. Fora, bE Q; we write

a = pcxa', b = pPb', a', b' E UQp.

If p #- 2, then

( a,b) _ p-lcxP(a,)p(b')CX - _ (-1) 2 - - • p p p

In particular, one has (P/) = (-I)(P-O/2 and(P/) = (*),ifuisaunit.

Ifp = 2 anda,b E Uo.2' then

(2; a) = (_1)(a2-1)/8 ,

(a;b) = (b;a) = (-1) a2l b2l .

Proof: The claim for the case p #- 2 is an immediate consequence of (3.4),

and will be left to the reader. So let p = 2. We put .,,(a) = a2 ~ 1 and

sea) = a; 1. An elementary computation shows that

.,,(ala2) == .,,(al) + .,,(a2) mod 2 and s(ala2) == s(al) + s(a2) mod 2.

Thus both sides of the equations we have to prove are multiplicative and it is enough to check the claim for a set of generators of UQ2/Uij2. {5, - I}

is such a set. We postpone this for the moment and define (a, b) = (a:/).

338 Chapter V. Local Class Field Theory

We have (-I,x) = I if and only if x is a nonn from «:l!2(R)I«:l!2' i.e., x = y2 + z2, y,z E «:l!2. Since 5 = 4 + I and 2 = 1 + 1, we find that ( -1, 2) = (-1, 5) = 1. If we had (-1, - 1) = 1, then it would follow that ( -1, x) = 1 for all x, i.e., -1 would be a square in «:l!~, which is not the case. Therefore we have (-1, - 1) = -1.

We have (2,2) = (2, - 1) = 1 and (5,5) = (5, - 1) = 1. It remains therefore to detennine (2, 5). (2,5) = 1 would imply (2, x) = I for all x, i.e., 2 would be a square in «:l!~, which is not the case. Hence (2, 5) = -1.

By direct verification one sees that the values we just found coincide with those of (_I)IJ(a), resp. (_l)e(a)e(b), in the respective cases.

It remains to show that UQ2/Uij2 is generated by {5, - I}. We set U =

UQ2' u(n) = U~~. By chap. II, (5.5), exp : 2n Z 2 --+ u(n) is an isomorphism

for n > 1. Since a t-+ 2a defines an isomorphism 22Z 2 --+ 23Z 2, X t-+ x 2

defines an isomorphism U(2) --+ U(3). It follows that U(3) ~ U2. Since {I, - 1, 5, - 5} is a system of representatives of U / U (3) , U / U2 is generated by -1 and 5. 0

It is much more difficult to detennine the n -th Hilbert symbol in the general case. It was discovered only in 1964 by the mathematician HELMUT

BRUCKNER. Since the result has not previously been published in an easily accessible place, we state it here without proof for the case n = pV of odd residue characteristic p of K.

So let /Lp" ~ K, choose a prime element rr of K, and let W be the ring of integers of the maximal unramified subextension T of K I«:l!p (i.e., the ring of Witt vectors over the residue class field of K). Then every element x E K can be written in the fonn

x = f(rr),

with a Laurent series f(X) E W((X».

For an arbitrary Laurent series f(X) = LiO':-m aiXi E W((X», let fP (X) denote the series

fP(X) = LarXip , i

where cp is the Frobenius automorphism of W. Further, let Res(f dX) E W denote the residue of the differential f dX,

f' dlogf := fdX,

and logf := f(_l)i+l (f ~ l)i ,

i=l I

if f E 1 + pW[[X]].

§ 3. The Hilbert Symbol 339

Now let { be a primitive pV -th root of unity. Then 1 - { is a prime element of Ql p ((), and thus

for some unit c of K, where e is the ramification index of K IQlp({)' Let 1}(X) E W[[X]] be a power series such that

c = 1}(n),

and let h (X) be the series

1 1 + (1 - xe1}(X))pv heX) - - -----....,,­

- 2 1 - (1 - xe1}(X))pV

00

L ai Xi , ai E W, i=-oo

lim ai = O. i---+-oo

With this notation we can now state BRUCKNER'S formula for the pV -th Hilbert symbol (xi/)' p = char(K) =F 2.

(3.7) Theorem. Ifx,y E K* and f,g E W«X))* such that fen) = x and g(n) = y, then

(x/) = (w(x,y)

where

( 1 fP 1 gP 1 w(x, y) = Trwlzp Resh· -log --:pdlogg- -log p -dlogf'P) modpv.

p f p g p

For the proof of this theorem, we have to refer to [20] (see also [69] and [135]). BRUCKNER has also deduced an explicit formula for the case n = 2v , but it is much more complicated. A more recent treatment of the theorem, which also includes the case n = 2v , has been given by G. HENNIART [69].

It would be interesting to deduce from these formulae the following classical result of !WASAWA [80], ARTIN and HASSE (see [9]) relative to the field

<PV = Qlp({),

where { is a primitive pV -th root of unity (p =F 2). Putting n = 1 - { and denoting by S the trace map from <Pv to Qlp' we obtain for the pV -th Hilbert

symbol (X i/) of the field <Pv the

340 Chapter V. Local Class Field Theory

(2 v-I) (3.8) Proposition. For a E U <1>: and b E <P~ one has

(1) (a~b) = ~S(l;logaDlogb)/pV,

where D log b denotes the fonnalJogarithmic derivative in rr of an arbitrary representation of b as an integral power series in rr with coefficients in Z p'

For a E U~'! one has furthennore the two supplementary theorems

(2) (~ ~ a) = ~S(Ioga)/pv,

(3) (a~ rr) = ~S((Nn) loga)/ p" .

The supplementary theorems (2) and (3) go back to ARTIN and HASSE [9]. The fonnula (1) was proved independently by ARTlN [10] and HASSE [61] in the case v = 1, and by !WASAWA [80] in general. In the case v = 1, for instance, one can indeed obtain the fonnulae from BRUCKNER'S theorem (3.7). Since

-S(~rrl) == 1 . {1 mod p,

p 0 mod p,

i=p-1,

i ::f= p - 1, and loga == 0 mod p2,

one may also interpret the l; -exponent in the formulae (1)-(3) as the (p -1)­st coefficient of a rr -adic expansion of log a D log b. In this way it appears as

a fonnal residue Resn ;p log a D log b. As to the supplementary theorems,

one has to define also Dlog~ = -~-', Dlogrr = rr-'.

Exercise 1. For n = 2 the Hilbert symbol has the following concrete meaning:

( a, b) = 1 ~ ax2 + bi - Z2 = 0 has a nontrivial solution in K. P

Exercise 2. Deduce proposition (3.8) from theorem (3.7).

Exercise 3. Let K be a local field of characteristic p, let K be its separable closure, and let Wn(K) be the ring of Witt vectors of length n, with the operator tp: WnCK) -+ Wn(K), tpa = Fa - a (see chap. IV, §3, exercises 2 and 3). Show that one has ker(tp) = W n (IF p).

Exercise 4. Abstract Kummer theory (chap. IV, (3.3» yields for the maximal abelian extension L I K of exponent n a surjective homomorphism

Wn(K) -+ Hom(G(LIK), Wn(lF p», x 1-+ Xx,

where one has Xx(a) = a~ - ~ for all a E G(LIK), with an arbitrary ~ E Wn(L) such that tp~ = x. Show that x 1-+ Xx has kernel tpWn(K).

§ 4. Fonnal Groups

Exercise 5. Define, for x E Wn(K) and a E K*, the symbol [x,a) E WnOFp) by

[x,a):= xA(a,LJK»,

where ( ,L JK) is the nonn residue symbol. Show:

(i) [x,a) = (a,K(~)JK)~ -~, if ~ E Wn(.K) with gJ~ =X.

(ii) [x + y,a) = [x,a) + [y,a).

(iii) [x,ab) = [x,a) + [x,b).

341

(iv) [x,a) = 0 {::::=} a E NK(~)IKK(~)*, where ~ E WnCK) is an element such that gJg = x.

(v) [x,a) = 0 for all a E K* {::::=} x E gJWn(K).

(vi) [x,a) = 0 for all x E Wn(K) {::::=} a E K*pn.

Exercise 6. Let K be the residue class field of K and rr a prime element such that K = K«rr». Let

d : K -+ ilh, f ~ df,

be the canonical map to the differential module of KJK (see chap. III, §2, p.2(0). For every f E K one has

df = f;drr,

where f; is the fonnal derivative of f in the expansion according to powers of rr with coefficients in K. Show that for w = (Li>-OO airri)drr, the residue

Resw:=a_l

does not depend on the choice of the prime element rr.

Exercise 7. Show that in the case n = 1 the symbol [x,a) is given by

da [x,a) = TrKIIFp Res{x-).

a

Remark: Such a fonnula can also be given for n :::: 1 (P. KOLCZE [88]).

§ 4. Formal Groups

The most explicit realization of local class field theory we have encoun­tered for the case of cyclotomic fields over the field Q p' i.e., with the ex­tensions QpCOIQp' where S is a pn_th root of unity. The notion of formal group allows us to construct such an explicit cyclotomic theory over an ar­bitrary local field K by introducing a new kind of roots of unity which are "division points" that do the same for the field K as the pn -th roots of unity do for the field Q p'

342 Chapter V. Local Class Field Theory

(4.1) Definiton. A (1-dimensional, commutative) formal group over a ring 0

is a fonnal power series F (X, y) E o [[X ,y]] with the following properties:

(i) F(X, Y) == X + Y mod deg2,

(ii) F (X ,y) = F (Y , X) "commutativity",

(iii) F(X, F(Y, Z» = F(F(X, y), Z) "associativity".

From a fonnal group one gets an ordinary group by evaluating in a domain where the power series converge. If for instance 0 is a complete valuation ring and p its maximal ideal, then the operation

x + Y := F (x, y) F

defines a new structure of abelian group on the set p.

Examples:

1. Ga(X, Y) = X + Y (the fonnal additive group).

2. Gm(X, Y) = X + Y + XY (the fonnal multiplicative group). Since

X + Y + XY = (1 + X)(l + Y) - 1,

we have (x + y) + 1 = (x + 1) . (y + 1).

IGm

So the new operation + is obtained from multiplication . via the translation IGm

x~x+1.

3. A power series I(X) = alX + a2X2 + ... E o[[X]] whose first coefficient al is a unit admits an "inverse", i.e., there exists a power series

1-1(X) = at l X + ... E o[[X]],

such that 1-1 (f(X» = 1(f-l(X» = X. For every such power series,

F (X ,y) = I-I ( I (X) + I (Y) )

is a fonnal group.

(4.2) Definition. A homomorphism I : F ~ G between two fonnal groups is a power series I (X) = al X + a2X2 + ... E o[[X]] such that

I(F(X, Y» = G(/(X), I(Y») .

§ 4. FOITllal Groups 343

In example 3, for instance, the power series I is a homomorphism of the formal group F to the additive group Ga. It is called the logarithm of F.

A homomorphism I : F -+ G is an isomorphism if at = f' (0) is a unit, i.e., if there is a homomorphism g = 1-1 : G -+ F such that

l(g(X)) = g(/(X)) = X.

If the power series I(X) = atX + a2X2 + ... satisfies the equation I(F(X, Y)) = G(f(X), I(Y)), but its coefficients belong to an extension ring d, then we call this a homomorphism defined over d. The following proposition is immediately evident.

(4.3) Proposition. The homomorphisms I : F -+ F of a formal group F over 0 form a ring Endo(F) in which addition and multiplication are defined

by

(f + g)(X) = F(/(X),g(X)) , (f 0 g)(X) = l(g(X)). F

(4.4) Definition. A formal 0 -module is a formal group F over 0 together with a ring homomorphism

o -+ Endo(F) , a 1---+ [a]F(X) ,

such that [a]F(X) == aX mod deg 2. A homomorphism (over d ;2 0) between formal 0 -modules F, G is a

homomorphism I : F -+ G of formal groups (over d) in the sense of (4.2) such that

l(ra]F(X)) = [a]a(/(X)) for all a E 0.

Now let 0 = OK be the valuation ring of a local field K, and write q = (OK: P K ). We consider the following special formal oK-modules.

(4.5) Definition. A Lubin-Tate module over 0 K for the prime element 11: is a formal oK-module F such that

[1I:]F (X) == xq mod 11: •

344 Chapter V. Local Class Field Theory

This definition reflects once more the dominating principle of class field theory, to the effect that prime elements correspond to Frobenius elements. In fact, if we reduce the coefficients of some formal o-module F modulo Jr, we obtain a formal group P (X, Y) over the residue class field IF q. The reduction mod Jr of [Jr]F (X) is an endomorphism of P. But on the other hand, I(X) = xq is clearly an endomorphism of P, its Frobenius endomorphism. Thus F is a Lubin-Tate module if the endomorphism defined by a prime element Jr gives via reduction the Frobenius endomorphism of P.

Example: The formal multiplicative group Gm is a formal Zp-module with respect to the mapping

00

Zp ~ Endzp(Gm ), a 1--+ [a]lGm(X) = (l +X)a -1 = L e)Xv. v=l

Gm is a Lubin-Tate module for the prime element p because

[p]lGm(X) = (1 + X)P - 1 == xP mod p.

The following theorem gives a complete and explicit overall view of the totality of all Lubin-Tate modules. Let e(X), e(X) E OK [[X]] be Lubin-Tate series for the prime element Jr of K, and let

Fe(X, Y) E OK [[X, Y]] and [a]e,e(X) E OK [[X]]

(a E OK) be the power series (uniquely determined according to (2.2)) such that

Fe(X, Y) == X + Y mod deg2, e( Fe(X, Y») = Fe( e(X), e(Y») ,

[a]e,e(X) == aX mod deg2, e([a]e,e(X)) = [a]e,e(e(X)).

If e(X) = e(X) we simply write [a]e,e(X) = [a]e(X).

(4.6) Theorem. (i) The Lubin-Tate modules for Jr are precisely the series Fe (X , y), with the formal oK-module structure given by

OK ~ EndoK (Fe), a ~ [a]e(X).

(ii) For every a E OK the power series [a]e,e(X) is a homomorphism

[a]e,e : Fe ~ Fe

of formal 0 -modules, and it is an isomorphism if a is a unit.

§ 4. Fonnal Groups 345

Proof: If F is any Lubin-Tate module, then e(X) := [rr]F(X) E ETC and F = Fe because e(F(X,y)) = F(e(X),e(y)), and because of the uniqueness statement of (2.2). For the other claims of the theorem one has to show the following formulae.

(1) Fe(X,Y) = Fe(Y,X),

(2) Fe (X , Fe(Y, Z)) = Fe (Fe (X ,y), Z),

(3) [a]e,e(Fe(X, Y)) = Fe([a]e,e(X), [a]e,e(Y)),

(4) [a + b]e,e(X) = Fe([a]e,e(X), [b]e,e(X)),

(5) [ab] =(X) = [ale e([b]- =(X)), e,e 'e,e

(6) [rr]e(X) = e(X).

(1) and (2) show that Fe is a formal group. (3), (4), and (5) show that

OK ----+ EndoK(Fe), a t----+ [ale,

is a homomorphism of rings, i.e., that Fe is a formal oK-module, and that [a]e,e is a homomorphism of formal oK-modules from Fe to Fe. Finally, (6) shows that Fe is a Lubin-Tate module.

The proofs of these formulae all follow the same pattern. One checks that both sides of each formula are solutions of the same problem of (2.2), and then deduces their equality from the uniqueness statement. In (6) for instance, both power series commence with the linear form rr X and satisfy the condition e([rr]e(X)) = [rr]e(e(X)), resp. e(e(X)) = e(e(X)). 0

Exercise 1. Endo(<Ga ) consists of all aX such that a E o.

Exercise 2. Let R be a commutative Q-algebra. Then for every fonnal group F (X, Y) over R, there exists a unique isomorphism

logF: F ~ <Ga ,

such that logF (X) == X mod deg 2, the logarithm of F.

Hint: Let FI = aFjay. Differentiating F(F(X,y),Z) = F(X,F(Y,Z)) yields FI (X, 0) == 1 mod deg 1. Let 1jf(X) = 1 + 2::1 anxn E R[[X]] be the power series such that 1jf(X)FI (X,O) = 1. Then logF(X) = X + 2::1 ~xn does what we want.

00 xn Exercise 3. loglGm (X) = 2: (_1)n+1 n = logO + X).

n=1

Exercise 4. Let rr be a prime element of the local field K, and let ! (X) X + rr- I xq + rr-2 Xq2 + ... Then

F(X,Y) = rl(f(X) + !(Y)) , [a]F(X) = r\a!(X)) , aEOK,

defines a Lubin-Tate module with logarithm logF = !.

346 Chapter V. Local Class Field Theory

Exercise 5. Two Lubin-Tate modules over the valuation ring OK of a local field K, but for different prime elements rr and jf, are never isomorphic.

Exercise 6. Two Lubin-Tate modules.!e and Fe for prime elements rr and jf always become iso!,llorphic over oj(, where K is the completion of the maximal unramified

extension K I K .

Hint: The power series e of (2.3) yields an isomorphism e : Fe ---+ Fe.

§ 5. Generalized Cyclotomic Theory

Formal groups are relevant for local class field theory in that they allow us to construct a perfect analogue of the theory of the pn -th cyclotomic field Qp(S) over Qp, with its fundamental isomorphism

(see chap. II (7.13)), replacing Qp by an arbitrary local ground field K. The formal groups furnish a generalization of the notion of pn -th root of unity, and provide an explicit version of the local reciprocity law in the corresponding extensions.

A formal a K -module gives rise to an ordinary a K -module if we read the power series over a domain in which they converge. We now choose for this the maximal ideal p of the valuation ring of the algebraic closure j{ of the given local field K. If G(X I, ... , Xn) E aK [[XI, ... , Xn]] is a power series with constant coefficient 0, and if XI, ... , Xn E p, then the series G(XI, ... ,xn) converges in the complete field K(XI, ... ,xn) to an element in p. From the definition of the formal a-modules and their homomorphisms we therefore obtain immediately the

(5.1) Proposition. Let F be a formal a K -module. Then the set p with the operations

X+y = F(x,y) and a'x = [a]F(x), F

X, YEP, a E a K , is an a K -module in the usual sense. We denote it by P F'

§ 5. Generalized Cyclotomic Theory 347

If I: F -+ Gis a homomorphism (isomorphism) offormal oK-modules, then

I : PF -+ PG' x t---+ I(x),

is a homomorphism (isomorphism) of ordinary oK-modules. The operations in P F, and particularly scalar multiplication a • x =

[a] F (x), must of course not be confused with the usual operations in the field K.

We now consider a Lubin-Tate module F for the prime element n of OK.

We define the group of nn-division points by

This is an oK-module, and an OK Inn 0 K -module because it is killed by nn OK .

(5.2) Proposition. F(n) is a free OK /nn OK -module of rank 1.

Proof: An isomorphism I : F -+ G of Lubin-Tate modules obviously induces isomorphisms I : PF -+ PG and I : F(n) -+ G(n) of OK­modules. By (4.6), Lubin-Tate modules for the same prime element n are all isomorphic. We may therefore assume that F = Fe, with e(X) = xq +n X = [n]F(X). F(n) then consists of the qn zeroes of the iterated polynomial en(X) = (e 0 ••• 0 e)(X) = [nn]F(X), which is easily shown, by induction on n, to be separable. Now if A.n E F(n) "- F(n - 1), then

OK -+ F(n), a t---+ a· A.n ,

is a homomorphism of oK-modules with kernel nn 0 K . It induces a bijective homomorphism oK/nnoK -+ F(n) because both sides are of order qn. 0

(5.3) Corollary. Associating a f-+ [a]F we obtain canonical isomorphisms

Proof: The map on the left is an isomorphism since OK /nn OK ~ F(n) and EndoK (OK /nn OK ) = OK /nn OK . The one on the right is obtained by taking the unit groups of these rings. 0

348 Chapter V. Local Class Field Theory

Given a Lubin-Tate module F for the prime element 1'(, we now define the field of 7r" -division points by

Ln = K(F(n»).

Since F(n) ~ F(n + 1) we get a tower of fields

00

K ~ LI ~ L2 ~ ... ~ Lrr := U Ln. n=1

These fields are also called the Lubin-Tate extensions. They only depend on the prime element 1'(, not on the Lubin-Tate module F. For if G is another Lubin-Tate module for 1'(, then by (4.6), there is an isomorphism f : F ~ G, f E OK[[X]] such that G(n) = f(F(n» ~ K(F(n», and hence K(G(n» = K(F(n». If F is the Lubin-Tate module Fe belonging to a Lubin-Tate polynomial e(X) E err, then e(X) = [1l']F(X) and LnlK is the splitting field of the n-fold iteration

en(X) = (e 0 ... 0 e)(X) = [1'(n]F(X).

Example: If OK = 7l..p and F is the Lubin-Tate module Gm , then

en (X) = [pn]lGm (X) = (1 + X)pn - 1.

So Gm (n) consists of the elements ~ -1, where ~ varies over the pn -th roots of unity. Ln IK is therefore the pn_th cyclotomic extension Qp(/Lpn)IQp. The following theorem shows the complete analogy of Lubin-Tate extensions with cyclotomic fields.

(5.4) Theorem. L n I K is a totally ramified abelian extension of degreeqn-I (q-1) with Galois group

G(LnI K ) ~ AutoK(F(n») ~ UK/Ur> ,

i.e., for every (1 E G(Ln IK) there is a unique class U mod uj;>, with U E UK such that

AU = [U]F(A) for A E F(n).

Furthermore the following is true: let F be the Lubin-Tate module Fe associated to the polynomial e(X) E err, and let An E F(n) " F(n - 1). Then An is a prime element of L n, i.e., Ln = K(An), and

en (X) n-J ( 1) lPn(X) = = xq q- + ... + 1'( E OK[X]

en-I (X)

is its minimal polynomial. In particular one has NLnIK(-An) = 1'(.

§ 5. Generalized Cyclotomic Theory

Proof: If

is a Lubin-Tate polynomial, then

en(X) ¢n(X) = en-leX)

= en- l (x)q-l + rr (aq_Ien- 1 (X)q-2 + ... + a2en-1 (X)) + rr

349

is an Eisenstein polynomial of degree qn-l(q - 1). If F is the Lubin-Tate module associated to e, and An E F(n) " F(n -1), then An is clearly a zero of this Eisenstein polynomial, and is therefore a prime element of the totally ramified extension K(An)IK of degree qn-l(q - 1). Each (j E G(LIK) induces an automorphism of F(n). We therefore obtain a homomorphism

It is injective because Ln is generated by F(n), and it is surjective because

This proves the theorem. o

Generalizing the explicit norm residue symbol of the cyclotomic fields Qp(fLpn) IQp (see (2.4)), we obtain the following explicit formula for the norm residue symbol of the Lubin-Tate extensions.

(5.5) Theorem. For the field Ln IK of rrn-division points and for a = urrVK(a) E K*, u E UK, one has

(a, LnIK)A = [U-I]F(A) , A E F(n).

Proof: The proof is the same as that of (2.4). Let (j E G(Ln IK) be the automorphism such that

A(J = [U-I]F(A), A E F(n).

Let 6 be an element in Frob(Ln IK) such that (j = 61L and dK (6) = 1. We view 6 as an automorphism of the completion Ln = inK of Ln. Let E be the fixed field of 6. Since f ElK = d K ( 6) = 1, ElK is totally ramified. It has degree qn-l Vi -lJ because En K = K and 13 = E K = Ln. Consequently [E : K] = [Ln : K] = [Ln : K].

350 Chapter V. Local Class Field Theory

Now let e E Err, e E Eif be Lubin-Tate series over OK, where 1T = u 1T , and let F = Fe. By (2.3), there exists a power series e(X) = eX + ... E

0K[[X]], with e E UK' such that

eCP = e 0 [U]F and eCP 0 e = eo e (cp = CPK).

Let An E F(n) " F(n - 1). An is a prime element of L n, and

1TIJ = e(An)

is a prime element of E because

1T~ = eCP(A~) = ecp([U-1]F(An») = e(An) = 1TIJ.

'Since ei(e(An» = ecpi(ei(An)) = 0 for i = n, and =I- 0 for i = n -1, we have 1TIJ E Fe(n) " Fe(n - 1). Hence E = K(1TIJ) is the field of 1Tn_ division points of Fe, and NIJIK(-1TIJ) = 1T = un by (5.4). Since n = NLnIKC-An) E NLnIKL~, we get

rLnIK(a) = NIJIK(-1TIJ) = 1T == U mod NLnIKL~,

and thus

(5.6) Corollary. The field L n I K of1Tn -division points is the class field relative to the group (1T) x Ur-) S;; K*.

Proof: For a = U1T vK (a) we have

a E NLnIKL~ {:=:} (a,LnIK) = 1 {:=:} [U-t]F(A) = A for all A E F(n)

{:=:} [U-t]F = idF(n) {:=:} u-1 E Ukn) {:=:} a E (1T) X Ukn).

o

For the maximal abelian extension K ab I K, this gives the following generalization of the local Kronecker-Weber theorem (1.9):

(5.7) Corollary. The maximal abelian extension of K is the composite

K ab = KL rr ,

where L rr is the union U:' 1 L n of the fields L n of 1T n -division points.

§ 5. Generalized Cyclotomic Theory 351

Proof: Let L I K be a finite abelian extension. Then we have rr lEN L I K L * for suitable f. Since N L I K L * is open in K * , and since the U r) form a basis

of neighbourhoods of l, we have (rr/) x ur) S; N LIK L * for a suitable

n. Hence L is contained in the class field of the group (rr/) x ur) = «rr) x ur» n «rr/) x UK). The class field of (rr) x ur) is L n , and that of (rr I) x UK js the unramified extension K I of degree f. It follows that L S; KILn S; KLn = Kab. 0

Exercise 1. Let F = Fe be the Lubin-Tate module for the Lubin-Tate series e E Err, with the endomorphisms [a] = [ale. Let S = od[X]] and S* = {g E S I g(O) E Ud. Show:

(i) If g E S is a power series such that g(F(1» = 0, then g is divisible by [n], i.e., g(X) = [n](X)h(X), heX) E S.

(ii) Let g E S be a power series such that

g(X +A) = g(X) for all A E F(1), F

where we write X + A = F (X, A). Then there exists a unique power series heX) in F

S such that g=hon.

Exercise 2. If heX) is a power series in S, then the power series

h1(X)= Il h(X+A) AEF(l) F

also belongs to S, and one has h1(X +A) = h1(X) for all A E F(l). F

Exercise 3. Let N(h) E S be the power series (uniquely determined by exercise 1 and exercise 2) such that

N(h)o[n]= Il h(X+A). AEF(I) F

The mapping N : S ---+ S is called Coleman's norm operator. Show:

(i) N(h1h2) = N(hl)N(h2)'

(ii) N(h) == h mod p.

(iii) hE XiS' for i :::: 0 =} N(h) E XiS'.

(iv) h == 1 mod pi for i :::: I =} N(h) == I mod pi+l.

(v) For the operators NO(h) = h, Nn(h) = N(Nn-l(h», one has

W(h)o[nn]= Il h(X+A), n::::O. AEF(n) F

(vi) If hE XiS', i:::: 0, then Nn+l(h)/Nn(h) E S' and

Nn+1(h) == Nn(h) mod pn+l, n:::: O.

352 Chapter V. Local Class Field Theory

Exercise 4. Let A E F(n + 1) " F(n), n ~ 0, and Ai = [:7rn- i](A) E F(i + 1) for 0 ~ i ~ n. Then Ai is a prime element of the Lubin-Tate extension L H , = K(F(i + 1)), and 0i+' = odA;] is the valuation ring of Li+l. with maximal ideal PH' = AiOi+'. Show:

let fJi E :7rn- i p, 0i+', 0 ~ i ~ n. Then there exists a power series h(X) E S such that

h(Ai) = fJi for 0 ~ i ~ n.

Hint: Write fJi = :7rn- iAOhi (Ai), with hi(X) E o[X] and put, for 0 < ~ n: gi(X) = [:7rn+l][:7r i]/[:7ri+']. Then h = 'L7=ohigi is a solution.

Exercise 5. Let A E F(n + 1) " F(n) and Ai = [:7rn- i](A), 0 ~ i ~ n. For every U E ULn+l' there exists a power series h(X) E o[[X]] such that

Nn,i(U)=h(Ai) for O~i~n,

where Nn,i is the norm from L n+, to Li+l'

Hint: Write U = h,(A), h,(X) E o[X], and put h2 = Nn(h,) E S*. Show that fJi = Nn,i(U) - h2(Ai) E :7rn- ip,Oi+l' Then by exercise 4 there is a power series h3(X) E o[[X]] such that fJi = h3(Ai), 0 ~ i ~ n. Show that h = h2 + h3 works.

Remark: The solutions of these exercises are discussed in detail in [79], 5.2.

§ 6. Higher Ramification Groups

Considering the homomorphism

( ,L IK) : K* ---+ G(L IK)

defined for an abelian extension L I K of local fields by the norm residue symbol, it is striking that both groups are equipped with a canonical filtration: in the group K* on the left we have the descending chain

(*) K* ;2 UK = uf) ;2 u~) ;2 uf) ;2 ...

of higher unit groups U~), and on the right there is the descending chain

(**) G(LIK) ;2 GO(LIK) ;2 G 1(LIK) ;2 G2(LIK) ;2 .. ,

of ramification groups G i (L IK) in the upper numbering (see chap. II, § 10). The latter arose from the ramification groups in the lower numbering

Gi(LIK) = {u E G(LIK) I v£{ua - a) :::: i + 1 for all a E ad via the strictly increasing function

t dx l1LIK(S) = 10 (Go: Gx )

§ 6. Higher Ramification Groups 353

by the rule Gi(LIK) = G1frLIK(i)(LIK) ,

where 1/1' is the inverse function of TJ. We will now prove the remarkable arithmetic fact that the norm residue symbol ( ,L IK) relates both filtrations (*) and (**) in a precise way. To this end we determine (generalizing chap. II, § 10, exercise 1) the higher ramification groups of the Lubin-Tate extensions.

(6.1) Proposition. Let L n I K be the field of rrn -division points of a Lubin-Tate module for the prime element rr. Then

Gi(Ln IK) = G(Ln ILk) for l-J::::: i ::::: qk - 1.

Proof: By (5.4) and (5.5), the norm residue symbol gives an isomorphism

UK/uiP -+ G(LkIK) for every k. Hence G(LnILk) = (Ut),LnIK). We therefore have to show that

Gj(LnIK) = (Ut),LnI K ) for l-J::::: i::::: qk_1.

Let cr E GJ(LnIK) and cr = (u-J,LnIK). Then we have necessarily u E uf) because (,LnIK) : UK/Uj;) ~ G(LnIK) maps the p-Sylow

subgroup uf) / Ukn) onto the p-Sylow subgroup G J (Ln IK) of G(Ln IK).

Let u = 1 +srrm, s E UK, and)... E F(n) " F(n -1). Then)... is a prime element of Ln and from (5.4) we get that

)...0' = [U]F()...) = F()..., [srrm]F()...») .

If m :::: n, then cr = 1, so that VLn ()...O' - )...) = 00. If m < n, then )...n-m = [rrm]F()...) is a prime element of L n- m and therefore also [srrm]F()...) = [S]F ()...n-m). As Ln ILn- m is totally ramified of degree qm we may write [srrm]F()...) = so)...qm for some So E ULn • Since F(X,O) = X, F(O,y) = Y, we have F(X, Y) = X +Y +XYG(X, Y) with G(X, Y) E OK[[X, Y]]. Thus

i.e., {

qm if m < n, iLnIK(cr):=VLn()...O'-)...)= '.

00, Ifm::::n.

By chap. II, §1O, we have Gi(LnIK) = {cr E G(LnI K ) I iLnIK(cr) ::::

i + 1}. Now let qk-J ::::: i ::::: qk - 1. If U E ut), then m :::: k, i.e., iLnIK(cr) :::: qk :::: i + 1, and so cr E Gi(LnIK). This proves the inclusion

(ut),LnIK) £; Gj(LnIK). If conversely cr E Gi(LnIK) and cr =j:. 1, then

iLnIK(cr) = qm > i :::: qk-J, i.e., m :::: k. Consequently U E ut), and this

shows the inclusion Gi(Ln IK) £; (ut) , Ln IK). 0

354 Chapter V. Local Class Field Theory

From this proposition we get the following result, which may be consid­ered the main theorem of higher ramification theory.

(6.2) Theorem. If L IK is a finite abelian extension, then the nonn residue symbol

( ,L IK) : K* ---+ G(L IK)

maps the group uj;) onto the group Gn(L IK), for n ::: O.

Proof: We may assume that L I K is totally ramified. For if L ° I K is the maximal unramified subextension of L I K, then we have on the one hand Gn(LIK) = Gn(LILo) because o/LoIK(S) = S and o/LIK(S) = o/LILO(o/LOIK(S)) = o/LILO(S) (see chap. II, (10.8)). On the other hand, by chap. IV, (6.4), and chap. V, (1.2), we have

(Ui~),LILO) = (NLOIKUi~),LIK) = (Uj;),LIK) ,

so we may replace LIK by LILo.

If now L I K is totally ramified and Tr L is a prime element of L, then

Tr = NLIKCTrL) is a prime element of K and (Tr) X U~m) ~ NLIK L * for m

sufficiently big. Therefore L IK is contained in the class field of (Tr) X U~m), which, by (5.6), is equal to the field Lm of Trm-division points of some Lubin-Tate module for Tr. In view of chap. II, (10.9), and chap. IV, (6.4), we may even assume that L = Lm. By (6.1), the norm residue symbol maps the group ujF) onto the group

G(Lm ILn) = Gj(Lm IK) for qn-l ~ i ~ qn - 1.

But we have (see chap. II, § 10)

n 1 TJLIK(q -1) = -(gl + ... + gqn-l)

go

with gj = #Gj(L IK) = #G(LmILn) = (qm-l _qn-l)(q -1) for qn-l :::: i ::::

qn_1. This yields TJLIK(qn-1) = n and thus (UjF),LIK) = Gqn_l(LIK) = Gn(LIK). 0

Higher ramification groups G t (L IK) were introduced for arbitrary real numbers t ::: -1. Thus we may ask for which numbers they change. We call these numbers the jumps of the filtration {Gt(LIK)k:::_l of G(LIK). In other words, t is a jump if for all 8 > 0, one has

Gt(LIK) #- GtH'(LIK).

§ 6. Higher Ramification Groups 355

(6.3) Proposition (HASSE -ARF). For a finite abelian extension L I K , the jumps of the filtration {G t (L IK)}t::O:-l of G(L IK) are rationalintegers.

Proof: As in the proof of (6.2), we may assume (since G t (L IK) Gt(LILo)) that LIK is totally ramified and contained in a Lubin-Tate extension LmIK. If now tis a jump of {Gt(LIK)}, then by chap. II (10.9), t is also a jump of {G t (Lm IK)}. Since by (6.1), the jumps of {Gs(Lm IK)} are the numbers qn - 1, for n = 0, ... ,m - 1 (q = 2 is an exception: ° is not a jump), the jumps of {Gt(LmIK)} are the numbers f]LmIK(qn -1) = n, for n = 0, ... ,m - 1. 0

The theorem of HASSE-ARF has an important application to Artin L-series, which we will study in chap. VII (see chap. VII, (11.4)).