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A. S. Classification
A.8. Final Comments on Regularity of Haar Measures
4. Multiplicative Structure of Cp
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4. Multiplicative Structure of CP 147
hence ft - £2 =km is a multiple of in and
(gei-ez)
= q (gkm) _ `,m)k = (Zm)k = Zmk,
eP (xt)
_
(X2(x1)) = (g l- z) = z
zand finally
ze`(P(xi)=zezcp(XZ). U
Remarks. (1) For an additively written abelian group G, divisibility requires that all equations nx = a (x E G, n positive integer) have (at least) one solution x E G, hence the terminology. For example, the additive groups Q and R are divisible, but Z is not a divisible group.
(2) An abelian group G having the extension property mentioned in the statement of the theorem is called injective group or injective Z-module.
Application. The universal field Cp is algebraically closed; hence the multi- plicative group Cp is divisible. The homomorphism cp
: Z
Cp defined by cp(n) = p" E Cp has an extension 1/1 : Q --> Cp C. This extension is one-to-one, since its kernel is a subgroup of Q with ker 1/i fl Z = {0}. The image of i/r is a discrete subgroup I' C Cp isomorphic to the subgroup pQ c R,o. Instead of if(r) we shall often write pr E Cp and i/r(Q) = pQ. But - although the notation does not emphasize it - this subgroup pQ c Cp depends on a sequence of choices of roots of p in Cp and is not canonical. When we consider pQ as a subgroup of Cp,we have to remember that I p° I = I /p° > 0. This subgroup is a complement to
the kernel
U(1)=IX ECp:1XI=1}CCp
of the absolutevalue. In particular, we have a direct product decomposition
Cp=1.U(1)=poXU(1)
(analogous to polar coordinates in C") given by
x = r - (x/r) r-- (IxJ, x/r) (r E F, IxI _ Iri, x/r E U(1)).
Since
both AP and MP are clopen subsets of the metric space Cp, the subgroup U(l) = Ap -Mp is clopen and the preceding product is a topological isomorphism.
4.2 Roots of Unity
A firstanalysis of the structure of the group of units U(1) = Ap - Mp C Cp
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is made by looking at the reduction mod M. The restriction of the (ring) homo- morphism
e: AP-+ Ap/Mp=Fp-
(where Fpm is an algebraic closure of Zp/pZp = Fp) to units gives a surjective (group) homomorphism (11.4.3)
U(l) -> FP'-
with kernel E'(1) = I + MP C U(1), whence a canonical isomorphism
U(1)/(1 +Mp) = F.
In the algebraically closed field Cp, we can find roots of unity of all orders, so that it = tt(Cp) is isomorphic to the group of roots of unity in the complex field.
There is a canonical product decomposition of this group, It = FA.(p) . lip- (direct product),
where 1L p) is the subgroup consisting of the roots of unity of order prime to p, and /tpc the subgroup consisting of the pth power roots of unity (in Cp).
The restriction of the reduction homomorphism a gives an isomorphism of this subgroup µ(p) with FP'., and hence a direct product decomposition
U(1) = µ(p)
(1 +Mp) C C.
On the other hand,
lip-C (1+Mp)nQp.
Let us recall the more precise result established in (11.4.4).
Theorem. Let E lip- C Cp be a root of unity having order p` (t > 1). Then
I - ll = Iplt/wcd) < I (Op`) = p`-1(p - 1)) a
For a subextension K of Cp, the link with the notation used in (11.4.3) isµ(p)(K) = µ(p) n K: roots of unity (in K) having order prime to p, tt px (K) = it px fl K: pth power roots of unity (in K).
4.3. Fundamental Inequalities
In the preceding section (4.2) - based on 11.4.4 - we recalled the estimates for absolute values of pth powers. Such estimates form a recurring theme of p-adic analysis, and we give a few more precise forms of these estimates for convenient reference. The first one is purely algebraic.
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4. Multiplicative Structure of Cp 149
Fundamental Inequalities: First form. Denote by I = (p, T) the ideal of the ring Z[T] generated by the prime p and the indeterminate T. Then
(1+T)p' -1 ET In (n>0).
PROOF. For n = 0, the assertion is a tautology, and we proceed by induction on n > 0. Assume that (1 + T )P' = 1 + T u for some u E In. Hence
(I + T)p+' = (1 + Tu)p = 1 + pTuv + Tpup
for some polynomial v E Z[T]. ButpTuET-pI"CT-In+t,
Tpup=T - Tp-rup E T In+r
(since p > 2), and the sum pTu + Tpup belongs to T - In+r as expected.
Let us replace the indeterminate T by an element t E AP C CPI Since each element in 11 is a sum of terms containing factors p' T'-' for 0 < i < n, the ultrametric inequality shows that all elements obtained have an absolute value smaller than or equal to the maximum of I p'tn-` I, and we see that we have obtained the following inequality.
Fundamental Inequalities: Second f o r m. Let t c C p, l t I < 1. Then 1(1 +t)" - 11 < Itl - (max(Iti, Ipl))n (n > 0)
(cf. (V.4.3)).
Other forms are often used (they are not completely equivalent to the preceding ones, but also admit useful applications). We mention them briefly.
Third form. Let K bea finite extension of Qp, K D R D P. Then (I + P)p' C 1 + Pn+r (n > 0).
If P = xrR and IrrI = 0 < 1 (generator of the discrete group IKC R>o),
then in K the announced inclusion is equivalent to ItI <0 1(I+t)p° - 11 <Bn+r
This thirdform follows from the first one (replace T by -r) but is less precise than the secondform because
p E P, I pI = Be
and0=lplh/e>
IPlife> 1.
a.+
..s
Fourth form. With the some assumptions as in the third form, we have (1 + t)" m 1 + nt (mod pntR)
if t E 2pR (n E N, Z or even Zp).
If we look at the first term only in the expansion
(1 + t)" - 1 - nt = n(n - 1)t2/2+.--,
we find that for t/2 E pR,
n(n - 1)t2 t
2
=(n-1)
It only remains to check that the next terms are not competitive. Since we shall not need this form before Chapter VII, we refrain from giving a proof now. It will be obtained by a general method in (V.3.6).
4.4. Splitting by Roots of Unity of Order Prime to p
We have a direct product decomposition (4.2) U(1) = u(p) X (1 +Mp)
of the multiplicative subgroup defined by IxI =1 in C. The corresponding pro- jection U(l) -- p(p) is the Teichmuller character. It can be made explicit in several forms. Let lx i =1 and K = Q(x) have residue degree f. The residue field k = RIP of K has order q = pf, and the reduction homomorphism a sends the given unit x to an element e(x) E Fg of order dividing q - 1 (11.4.3). Hence
s(x)g-' = 1, xg t - 1
(mod P).The fundamental inequality (second form) shows that the pth powers of xg-t =
1 +r (t E PC K ort
M,, C C) tend to 1:X(q-1)p" -* 1 (n -* cc).
A fortiori, taking n = f m,
Xq'+1
Xqm = X(q-I)qm -p 1 (m __> cc).
Say xq'+'
= xq'(1 + sm) where Em -+ 0. Hence xq"' - xq' = xgmem - 0,
and the Cauchy sequence (xv" )m>o has a limit in the complete (locally compact) field K C Cp. Obviously, q = C and
= lim
m-->00Xq'=X+(x'
(.X,72-x')+----x (mod
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4. Multiplicative Structure of Cp 151 The map
x H = CJ(X) = llm
x9'm-cc
defines a homomorphism U(1) fl K" µq_1 C K" that corresponds to the projection on the first factor in the direct product decomposition (11.4.3)
U(1)f1K" =µq_1 x(1+P).
It is possible to give a formula working independently from the residue degree of x E U(1). Indeed, if q is given, the subsequence (xp") has an end tail in (xqm).
We have obtained the following result.
Theorem. Let X E CP with Ix I = 1. Then the sequence (xp"') converges to the unique root of unity that is congruent to x (mod MP) and the homomorphism
w:xr-> =a(x)= lim
m-* o0XP",corresponds to the projection on the first factor in the direct product decompo- sition
U(l) -
µ(p) x (I +Mp).4.5.
Divisibility of the Group of Units Congruent to I
In this sectionwe investigate the divisibility properties of the multiplicative group
I+Mp.
Proposition 1. The group 1 + MP is divisible. For eachm > 2 prime to p, it is uniquely m-divisible.
PROOF. It is enoughto prove that the group 1 + MP is p-divisible and uniquely m-divisible for eachm prime to p.
(1) Let 1 + tE 1 + MP and select a root x E Cp of XP - (1 +t): this is possible, since this fieldis algebraically closed. Since IX Ip = I xp I = I I + t I = 1, we have Ixl = 1:x E U(1). Now
(x mod Mp)p = xp mod Mp = I E k
implies x mod MP= 1. since k has characteristic p. This proves x = I + s E I + Mp.
(2) Let 1 + t E 1 + MP and select a positive integer m prime to p. We are looking fora root of the polynomial f (X) = X1 - (1 + t). We already have an
approximateroot y = 1 for which the derivative mXri-1 does not vanish mod MP (P doesnot divide m):
f(y) = 1 - (1 + t) = -t, f'(y) = m, If'(y)I = 1.
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Thus we have I f (y )/ f'(y)2 1 = I -t I < 1, and Hensel's lemma (11. 1.5) is applicable:
There is a unique root of f in the open ball of center I and radius 1. 0 In fact, for each E Um c µ(p) C FP'-, there is one root x of f with x (mod Mr). These m roots of f are all the roots of this polynomial, and for each given E ltm there can be only one root of f congruent to this root of unity
For later reference, let us formulate explicitly the following characterization of the topological torsion of Cp .
Proposition 2. For x E CP we have
xE1+MP
= XP,-> 1 (n --- oo).PROOF. If X = I + t E 1 + MP, the sequence
x''-I= (I+t)Pn-I
tends to 0 by the fundamental inequality (4.3) (second form). Conversely, assume that xP° -> 1 (for some x E Cp) and take an integer n such that xP' belongs to the open neighborhood 1 + MP of I in CP. Since we have proved in (4.1) that there is a torsion-free subgroup F (= pQ) of Cp and a direct-product decomposition
P
we see that X E p(p) . (1 +Mp ). The first component C of x is trivial simply because it has an order prime to p:
Observe that the convergent sequence is eventually constant precisely when x is a pth power root of unity