Math. Ann. 263, 509 514 (1983)

Am

9 Springer-Verlag1983

p-Ranks of Class Groups in Z~-Extensions Paul Monsky Department of Mathematics Brandeis University Waltham, MA 02254, USA

Introduction

Let L be a Galois extension of a number field k, with E = G(L/k) isomorphic to a product of d copies of the additive group of p-adic integers. Let k, be the fixed field of E p" and G, be the ideal-class group of k,. Let -~(L/k) be the dimension of the Z/p vector space GJpG,. We wish to study how ~,(L/k) grows with n. To state our results we need to introduce a certain module .~ over the d-variable power series ring with co-efficients in Z/p. Explicitly we let ML be the maximal abelian unramified extension of L whose Galois group is annihilated by p, and we set )~--G(f4L/L ). We take "4d to be the completion, with respect to powers of the augmentation ideal, of the Z/p group ring of E; it is isomorphic to Z/p[,[,X 1..... Xd] ]. )( has the structure of finitely generated (but not necessarily torsion) Ad-module ; in fact .~ is just the reduction mod p of the usual IwasawaGreenberg module as defined in [,1]. When )~ is a finite group we shall see that the "~(L/k) are constant for large n. When X is infinite the situation is more interesting. Let a be the altitude of the local ring Ad/AnnX, so that 1-< a-< d. We shall prove the following results: There is a real constant c > 0 such that ~,(L/k)= cp""+ 6,, where (~n.=O(p(a- 1),). When a=d, c is just the rank of Jr over eld. When a = l , c is an integer and 6, is a periodic function of n for large n. The proof uses [-1] to reduce to a module-theoretic problem which may be handled by the methods of our paper, [2], on the Hilbert-Kunz function. Various interesting and difficult questions remain untouched. What can be said about 6, when a > 1 ? When 1 < a < d, is the constant c rational ? And finally, do analogous results hold for the groups G./pJG. for fixed j > 1 and large n?

lo

We show how to calculate the "~.(L/k) from a knowledge of 3~ and some supplementary information about the inertia groups in ML/k. Our arguments closely parallel those made in Sects. 5 and 6 of [1]. We adopt the notation of the

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introduction. Let Y = G()C4L/k). Y maps onto E with kernel )(. Let ytn) C Y be the inverse image of E p" ; then Yt")= G(PlL/k.). For each prime of k which ramifies in L choose a prime of A4/, lying over it. Let J l , ..., J r C Y be the corresponding inertia groups and J j ~")= j j ~ Y~"). The ~,j /~.") are inertia groups in PlL/k,, and every inertia group in this extension is a conjugate of some ~ t~,l by some element of Y. J j maps bijectively on a Z : s u b m o d u l e j j of E. Now take al,...,adEY mapping on a basis of E. Let J r + j be the closed subgroup of Y generated by a~, and let :,~(n) r + ~ -__: ,~+' p "j .

Definition 1.1. C, is the closed subgroup oJ Y~") generated by [Yt"), y~,,I], all ffh powers oj elements oj Y~"), and all the inertia groups in G(ML/k,). B, is the intersection oJ C. with X. Using class-field theory, we find as in [1] : Lemma 1.2. The Z/p-dimension oj Y~")/C, is ~,(L/k). B, is a 71a-submodule oJ X,

and ]or n large, P,(L/k)-I(X/B,) is constant. Observe now that C, is the closed subgroup of Yt") generated by [Y~"), Y~")] and all the conjugates under Y of the ,r: j , 1 < j < r + d . So the situation we are in is virtually the same as that of Sect. 5 of [1] and we can define submodules A,, A', and A" of B, as in that section. In particular we find" Lemma 1.3. Let A, = IX, Y~")]. Then A, is a 71a-submodule oJ B,. In Jact, A, = I ~ where I, is the ideal generated by the (1 + X ) : - 1 in ~1d. Since p = 0 in -4n, I, is actually the ideal generated by the X~", or alternatively the ideal generated by all g : with g in the maximal ideal, m, of Ad" This enables us to use the results of [2] to estimate I(X/A,,). We next get information about B,/A,. r+d

Definition 1.4.

A; = A, + Z J]a[ Y,~ " ) ] 9 1

Lemma 1.5. A'~ =A', + [Y("), Y~")] is a 71d-SUbmodule oJ B,. For each n, (a, z ) ~ [ 6 , z] tt i tt t induces a Ze-bilinear map E p" x E p n ~ A,/A, whose image generates A,/A,. Thus the groups A"/A', have bounded order.

Lemma 1.6. Let J<")CJ~mx " ' " •162 consist of all (~1 ..... ~r+a) with H ~ = I o" r + d Then (~l . . . . . O ~ r + d ) - - ~ l ' O ~ 2 . . . ~ r + d induces a Zp-linear map of Jt") onto B,/A~. Thus the groups B,/A~ have bounded order. Lemma 1.7. Let ((9, m) be a local Noetherian Z/p-algebra oJ altitude a > 1 and M be pn a finitely generated (9-module. Let 1, be the ideal oJ (9 generated by all the x , x~m. Suppose that g6 m and that b is a fixed integer. Then l((ge"- b, I,)M/1,M)=O(p ~"- 1),). ProoJ. We may assume that M = (9. Let N C (9 = {x]x is annihilated by some power of g} and set (9' = (9/(N, gb). Since gb is not a zero-divisor on (_9/N, the altitude of el' is < a - 1. Now if m is generated by s elements, 1, 3 m ~p". We conclude that 1((9'/I,(9')
Theorem 1.8. Let (9=/]d/Ann)(, and a=altitude (9. Suppose that a > l . Then F.(L / k ) = I(X/ A,) + O(p~"- ~)").

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Pro@ "~,(L/k)= I(X/B,)+ constant for large n. In view of lemmas 1.5 and 1.6 it suffices to show that I(A',/A,)= O(p ~ - ~)"). Now there exists a v such that for n > v, -(J) ) . Using definition 1.4, and arguing as in the proof of Theorem 5.11 of [1] we find that A',/A, is generated as Ad-mOdule by finitely many elements ((~P"-1)/(~ p~- 1)).u where ~ and u are independent of n. Set g = ~ - 1 and b = p ~. Then (ffP"- 1)/(6p~- 1)=g p~ Lemma 1.7 then shows that A',/A, is a homomorphic image of a product of a fixed finite number of (9-modules, each of which has length O(p~"- a)). The theorem follows. Theorem 1.9. Notation as in Theorem 1.8. Suppose that a >=1. Then there is a real constant c > 0 such that "d,(L/k)= cpa" + 6, where ~ = O(p ta- 1),). When a = 1, c is an integer. When a = d, c is just the rank oj X over A d. ProoJ. The corollary to Theorem 1.8 of [2] shows that I(X/A,) = l ( X / I ~ ) = c(X)p ~" +O(p t~-')") with c0~)>0, and we apply Theorem 1.8 of this paper. When a = l , Theorem 1.10 of [2] shows that c(,~) is an integer. When a = d , Ann)s and C =.,ld. Let r be the rank of)~ as C-module. Then Lemma 1.3 of [2] shows that ~)~) = c(Cr). Since l(fld/I,)= pal,, c(cr)= r, completing the proof. When a = 1, Theorem 3.11 of [2] shows that l(X/A,)=cO~)p"+ (a periodic function of n) for large n. To show that a similar result holds for I(X/B.), and consequently for the "d,(L/k), is our final task. We shall assume until the last theorem of this section that the inertia groups in L / k are direct summands of E. Then of)") = JP" for 1 =
As in [1] we prove: Lemma 1.11. A', is generated as 71a-module by 1 ~ and finitely many j~,~(p") where the ~ and z do not depend on n. A',I is generated as An-module by A', and finitely many go, ~(P") where the a and z do not depend on n. B, is generated as 71d-module by A~ and jinitely many he(p" ) where the ~ejto) do not depend on n. We now abstract a property shared by the functions j~,~, g~,~ and hQ. Definition 1.12. Let (A, m) be a local Z/p-algebra. I~ g~ A we write gt.) .and g<,> to denote gP" and gP"- 1 respectively. I] g and h are in A we set (9, h), = ro'h J, the sum extending over all i and j with i + j = p " - 2 . Note that (9-h).(g,h),=g<,>-h<,>. Definition 1.13. A function from N to an A-module M is "speeiar' if it is of the jorm n ~ Z ( g i, hi),u i where the gi and h i are in m and the u i are in M. Lemma 1.14. View X as a module over 7td, or alternatively over (9 =,']d/AnnlY. Then the Junctions n~J~,~(p") oJ Lemma 1.11 are special Junctions N-~,~. Proof Suppose that ~r and z are in Y. Let g be the element ~ - 1 of the maximal /

ideal m of 71d. By Lemma 6.5 of [I], [ ~ , z ] =

~ ( t ) g J - a [ ~ , z ] . Taking l=p" we \J/ find that f~, ~(p")-- g<.>. [a, z] = (g, g),. ( - g[~, z]), proving the result. j=l

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Lemma 1.15. The Junctions n-~o,,~(P") and n-~ho(p" ) of Lemma 1.11 are special Junctions N-~ X,

Proof. We give the proof for the hQ ; the argument for the 9, ~ is the same. We adopt the notation of Lemma 6.6 of [1]. Then h~(l)=lu 1 + k ' ( l )kU k . So if n > l , k=2\

ho(p") = up,. But Lemma 6.6 of [1] shows that Up. =

~,

/

(Zj, Zj+ 1)n ~i~j[~i' Oj]

l~i
with the zjE m. The lemma follows. Combining the above results we find: Theorem 1.16. There is a finite set oJ special functions ~i : N ~ X such that, Jot

each n, B,, is the submodule oJ R generated by 1,, .X and the ~i(n). Remark. Suppose that 2~ is a finite group so that (9 =.4d/Ann)~ is Artinian. Then, for n large, I, =(0) and (g, h), = 0 in (9. So B, =(0) and Y,(L/k) is constant for large n. (There are simpler proofs of this, of course .) Section 2 will be devoted to a proof of the following variation on Theorem 3.11 of [23: Theorem 1.17. Let ((9, m) be a complete local Noetherian Z/p-algebra of altitude 1,

reduced, with finite residue class field and let I, be the ideal generated by the gP", gem. Let M be a finitely generated (9-module and ~ be finitely many special functions N-~M. Let B, be the submodule oJ M generated by I , M and the ~pi(n). Then I(M/B,)=c(M).p" +6,, where 6, is a periodic Junction oJ n for large n. The above results allow a precise description of the growth of the "F,(L/k) when a=l: Theorem 1.18. Suppose we are in the situation of Theorem 1.9 with a = 1. Then 6, is

a periodic Junction of n .[or large n. Proof. Replacing the Zap-extension L/k by L/k= for some s does not change 2( as a group. But its structure as a module over Z/p[[X1, ...,Xa] ] is changed, and the new Ann~" is precisely {g 19P=ethe old AnnX}. So if we replace k by k= with s large we may assume that AnnX is a radical ideal and that (9 is reduced. We may further assume that the inertia groups in G(L/k) are direct summands of E, so that the conclusion of Theorem 1.16 holds. Then Theorem 1.17 and Lemma 1.2 give the result.

.

This section is devoted to the proof of Theorem 1.17. Throughout (9, I,, M and B, are as in the statement of that theorem. Definition 2.1. s is the number oJ height 0 primes of (9 ; P1,'", P~ are those primes.

(9i is the integral closure oJ (9/Pi in its field oJ quotients, ord i is the valuation s

function in the complete discrete valuation ring (9r (9'= 1-] (gr 1

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Remark. Suppose that M is a finitely generated O-module. To say that M is a finite group is the same as saying that it has finite length over (9; that is to say that it is m-primary. This also amounts to the localization of M at each Pi being (0). Now let ~bl be the map (9--*(9~and ~b : (9--*(9' be the product of the ~b~.Since (9 is reduced, q~ is injective. Lemma 2.2. Suppose that ui6 (9~with each ordi(ui) large. Then u = (up..., us) is in the image oj (a.

ProoJ. We show first that the cokernel of q~ is m-primary. Since (9i is finite over 6/P~, (9' is a finite (9-module. So it suffices to show that the localization of 4) at each Pi is onto. But this is easy - the localizations of (9 and (9' at P~ are each just the field of quotients of (9/P~. It follows that cokernel ~b is annihilated by some h not in any Pi" Suppose now that ordi(ui) ~ ordi(h). (By ordi(h) we mean ordi(tpi(h)); we shall consistently abuse language in this way). Then ue h(9'C (image ~b), and we're done. Lemma 2.3. Suppose that (A,m) is a complete Noetherian local Z/p-algebra with ]inite residue class field F, o] degree r over Z/p. Suppose g and h are in A. Then as n~o~, n=a Jixed n o (r), the sequences gt,), g<,> and (g,h), converge in A.

Proof A is a homomorphic image of F [ [ X 1,...,Xd] ] for some d, and we can evidently replace A by that ring. If gem, gt.) and g<,>~0. Ifg~m, g =aft with ~ F and fl-:l(m). Then if n - n o (r), g~,~=:t~,)[l~,)=:~,o~[3{,~ and this -~<,ol as n--,oc ; if follows that 9<,>--*~{,o~.g-~. Suppose now that g.i=h. Since (g-h).(9, h), =g<,>-h<,>, we see from the above that ( g - h ) . ( g , h ) , ~ s o m e u in A. Since the ideal ( g - h ) A is closed, u = ( g - h ) (some v) and we see easily that the (g,h)--*v. Finally (g,g),= _ g p - - 2 and the argument given for g<,> shows that the (g,g), converge. Definition 2.4. ri is the Z/p-dimension oJ the residue class field o] (9i. r is the least common multiple o] the r i. We shall show that if r is taken as in Definition 2.4, then I(M/B,) = c(M)p" + 6, where 6, +, = 3, for large n. Until further notice we shall assume that the number of elements in the finite field (9/m is > the number of height 0 primes s, of (9. Lemma 2.5. Under the above assumption there exists an j 6 m such that ]or each i, ordi(J) = rain (ordi(g)). g~m

ProoJ. Let K C (9 be a co-efficient field projecting onto (9/m. Take an j ~ m such that ordi(J) =rain (ord~g) for a maximal subset J of the i's. gem

Suppose there is some index j such that j(EJ. Choose an h such that ord~(h) = rain (ordj(g)). For 2e K set h~ = h + 2J. Then ordi(h j = ordj(h). Furthermore for each i e J there is at most one 2 such that ord~(hj > o r d H ) . Since IKI>IJI we can find a 2 such that ordi(hj=ordi(]) for each i s J ; this contradicts the definition of J. Definition 2.6. With J as in Lemma 2.5 and r as in De]inition 2.4, we set F=]. (Note that F.J=J~r) and that F(,)"Jr,)=J~.+,)')

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Lemma 2.7. Suppose that g and h are in m and that n is large. Then: (a) F~.~. g~.) -= g~. + ~)m o d j~. + ~) (b) Ft.). (g, h). = (g, h). +~ m o d J(. +,).

Proof By our assumptions on J we m a y write g = ~r h = fl(ilj with c(i) and fl(i) in C i. (To simplify notation we write g for Oi(g), h for (oi(h)). Then, in C~, (0 -- ~(n+r)J' and u=(ul, "".,u~). Since F(n)" gin) _ g(n +r) _--Jtn +r)(~(n) n(1) ~ Let .ui-_ ~(i) ~(i) (.)--~(n+r) each rl divides r, L e m m a 2.3 applied to (9~ shows that ord~(u~) is large. By L e m m a 2.2, u=~b(x) for some x~C. Then Fr162 Similarly, (F(.).(g,h).) 1 1 - (g, h). +~= -f-s +r)((ati), fl(i~). _ (o~(i), fl(i)). + r) in C i. Setting u i -= (t~i(j)) 2 ((~(i), fl(i))n

_(at0,fl.~).+~) and arguing as above, using L e m m a s 2.3 and 2.2, gives the result. An immediate consequence is: Corollary. ( a ) S u p p o s e g~m, u~M. For n large, F~.).(g~.)u)=g~.+.~u m o d the submodule J~, +r~m oj m. (b) Let ~ : N ~ M be a special function. For n large, F(,) "Op(n)) - q~(n + r) m o d ft. + ~)M.

Lemma 2.8. For n large, l(Bn/jr

) = l(B,+,/J~,+r)M ).

Prooj. (j) is an m - p r i m a r y ideal; it follows that the a b o v e modules have finite length. So it suffices to construct an O-linear m a p of B,/~[!,)M onto B, +r/J~, +r)M for all large n. N o w multiplication by F~,) induces a m a p M/f(,)M--*M/f,+r)M. Let gCi)be generators of m and gj be generators of M. The image of B,/Ji,)M under the above m a p is the submodule of M/f~,+r~M generated by the F~,).(gl~,~uj) and the F~,).~pi(n ). But the above corollary shows this submodule to be precisely B,+,/J(,+r)M for large n. We shall now prove T h e o r e m 1.17 ; in fact we shall show that 6,+~ = 6 , for large n if r is chosen as in Definition 2.4. Suppose first that [O/m[ >=s, so that j m a y be chosen as in L e m m a 2.5. Since (j) is m-primary, I(M/jkM) is a polynomial of degree __<1 in k for large k. Taking k=p" and applying L e m m a 2.8 gives the result. The general case m a y be handled by an "extension of ground field" argument. Let K C (9 be a co-efficient field projecting onto (9/m. Let l be a large prime, K # the degree I extension of K, and C ~ the tensor product of C with K ~ over K. Then C ~ is complete local Noetherian of altitude 1, reduced, with residue class field K ~. We m a y assume that I does not divide r. It is then easily seen that the height 0 primes of C # are just the P/9 ~, and that the "r{' for this prime ideal is Iri. N o w let M ~ be the tensor product of M with C ~ over C and B~ be the C ~ - s u b m o d u l e of M ~ generated by I , M ~ and the ~Pi(n). Then the C-length of M / B , is just the C~-length of M ~ / B ~ . So by our above result, applied to ( ~ , I(M/B,)= c(M~)p"+ 6, where 6,+~ = 6 . for large n. Replacing l by a n o t h e r large prime l' we get the desired result.

References i. Cuoco, A., Monsky, P. : Class numbers in Z~-extensions. Math. Ann. 255, 235-258 (1981) 2. Monsky, P.: The Hilbert-Kunz function Received March 18, 1983