Zp-extensions of k, i.e. Galois extensions of koo/k with Galois group r__g r. Denote by k,~ the n O' layer of koo/k, so that k,~/k is a cyclic extension of degree p~.
m a n u s c r i p t a math. 62,
145 - 161 (1988)
m a n u s c rip ta m a t h e m a t i ca Springer-u
1988
On G e o m e t r i c Z f E x t e n s i o n s o f F u n c t i o n Fields R.GOLD AND H.KISILEVSKY 1
w
Introduction. In this paper we consider Zfextensions of a function field k over a finite field of
characteristic p. The constant gfextension of k has often provided a useful analogy for the study of Zfextensions of number fields. In this article we study the properties of geometric Zfextensions of k and compare them with corresponding properties in the other two cases. In w we examine the growth rate of the class number in a geometric Zp-extension and show that it is much faster than the growth of class numbers in either the number field case or in the case of constant extensions. In w we consider the p-part of the class group and the Iwasawa theory in this sitnation. Here we find that the Iwasawa modules that arise can be quite different from those one finds in the number field case. We also give some examples of Z~-extensions which have non-trivial invariants in the sense of Cuoco and Monsky. Some of these results were announced at the Laval meeting [G-K].
w
Function fields. Suppose that k is a function field in one variable with constant field Fq, (q = f ) , so
that there exists an t 6 k, transcendental over Fq such that [k : Fq(z)] < oo. We consider Zp-extensions of k, i.e. Galois extensions of koo/k with Galois group r__g r Denote by k,~ the n O' layer of koo/k, so that k,~/k is a cyclic extension of degree p~. lttesearch supported in part by grants from NSERC and FCAlq,
145
Gold
-
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Denote by D0(k) the group of divisor classes of k of degree 0, and by h(k) the order of ~Do(k), so that 2g h(k) = R ( 1
-
i=1 where - ai~)
I is the (-function of k and t = q - ' . The numbers el are algebraic integers with ~ti~i
=
q.
Let ~ denote the unramified Zp-extemion of k, i.e. the Zp-extension of/r obtained by adjoining to k the constant Zfextension Fq of Fq. For boo = k, 29
,(k.) = 1-l(t - a..") 0=1
It follows that for the C A layer of k / k we have h,(k~)
~, qgP"
as
~ ~
CO.
There is the following analogue of the Brauer-Siegel theorem due to Madan and Madden [M-Ma]. For function fields K with constant field Fg, log h ~ g log q
as
m / g ~ 0,
where m = mln=~K[K : Fq(m)]. We shall apply this result to non-constant Zfextemions in the following. Let koo/Ir be a geometric Zfextensiou (i.e. for all n, k,, has constant field Fq). Let /~,, = h(k,,) and 9,* = g(~:,,) be the class number and genus of/r
respectively, in order to
study the growth of h,, as u ~ CO we must investigate the growth rate of g,,. For this we recall the Riemann-Hurwitz genus formula: 2(9,~ - I) = 2p'~(g(k) - 1) -I- deg(DCl%/k))
146
Gold - K i s i l e v s k y where D(k,,/k) is the different ideal of the extension kn/k.
IfD(k,~/k) = "P~'... P~", where
:Pl,-.., :Pr are the primes of kn ramified over k , then since the ramification is wild we have
~i = m i ( p - 1)
with
mi _> 2.
Hence in order to obtain an asymptotic growth rate for h,~ as n --* or, we use class field theory to ealc~ate vC~./k ) and 9(k,). Let G = Oal(k"b/k.) be the Oalois group of the abelian closure k "b over k, and let H be the maximal abelian unramified extension of k (in k'b). Then there is a commutative diagram: 0
~
o
--,
v,t(~"/~r)
1
O.l)
~k*Ir
-~
c
~
Jklk'
.T
----,
vat(~/k)
---~
T
~(k)
--,
0
---,
0
where p : ,fk/k* ~ G is the reciprocity homomorphism of the id~Ie class group ff~/k* into G, U = 1] Up is the group of unit id~les of k (the product of the local unit groups Up), and ~(k) is the group of divisor classes of k. We recall that p is injective on ,fk/k* with dense image and induces an isomorphism
U~*lr ~- V/F*, ~_ G,,tCk'bln). By taking pro-p-primary parts in (1.1) (see w we obtain
o~
]-Iv~ - ~ o ,
- ~ 7,, • 90(k), - , o
where U~, is the group of 1-units of kT,, the completion of k at ~o and :Do(k)p (resp. Gp) is the p-primary part of D0(k) (resp. G). Choosing a uniformizing parameter ~r at 7~ we see that k~, _~ Fq,((Tr)), the field of Laurent series in 7r over FC, with d = deg('P) and that
U~ -- 1 + rrF,, [[r.]]
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Gold
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Kisilevsky
is a free abellan pro-p-group of infinite rank. If ~01,...,w, 9 k represent a basis of t] residue class field extension at 7>, Fqd/Fp, (p~ = qd), then we show that every ct 9 U~, h a unique expansion as a convergent infinite product of the form a = l ' I (1 - wiTrJ)=u where a~j 9 Zp, 1 < i < s, j > 1, and PXJ. REMARK 1 : Gp ~ Z~ ~ is a free abelian pro-p-group of infinite rank since it is the col mutator quotient of a free pro-p-group, the Galois group of the maximal p-extension k. PROPOSITION 1. If U 1 = 1"[ U~ then U 1 is a free abelian pro-p-group with topologh
b~sls {e,j(~')} where e~j('P) = :t - ,,,~,','~
:t < i < s, i > 1, P,I'i.
For any continuous homomorphJsm f : U 1 ~
g : Gp
Zp there exists a continuous homomorp~,~
, zp such that g(%) = f(e~j) for all but a ~nite number of ~ ( # ) .
PROOF: Since U t is a product of the groups U~ it is sufficient to prove that the elemel 1 - ~oi~'J are a free basis for the U~,. For this we must prove that they are independent a topologically generate U~,.
Suppose ~ = 1 + b ~ + . . . 9 k~, = Fp. [[~]1. W~ite j = j0p '~, p~'j0, and let b = b0p" 4'~(b0) e F , . , where 4 9
Gal(Fp./Fp) is the
Frobenius automorphism.
Then a ~ 1 + b0P'~rf j ~
_= (1 + b0~so),"
mod(Trj+l)
mod(~+ 1)
" C",)" n - ( 1 ~ (1 - ~,,,~,o)
mod(~ j+l)
i=1
where -b0 = Y~ clwl,
el E Z. Then al = aHe#
='P" - 1
148
mod(~ "i+t)
Gold
-
Kisilevsky
and proceeding by induction it follows that the eli topolo~cally generate U~. Suppose [I ei~~ = I. Let a 0 = p'~(id)ulj with ulj 6 Z;. If aq ~t 0 for some (ip j) choose (io,j0) so that J = jop '*(i~176is minimal. Since PXJ, we see ]pn(ij) = j,pn(r j') if and only i f j = j ' and n(i,j) = rt(i',j')= n. For all (i,j)
e.~' = (1 - ~,..J)'" =- 1 unless j
=
(rood
~"§
jo. In this case
II
~ - l'I
)
-
i=1
- 1 + ~U~jo~"~r "~ rood(# +~) i=1
1 mod(~"s+1) since w~" = ~'~(wl) are linearly independent over F~. To prove the second statement, suppose that G is a free abelian pro-p-group and that U C G is an open subgroup of index p. Let {e,,} be a topological basis for U and suppose that ~ E G, x ~ U. Then pz = ~ anen with a,, E Zp but p~'a,, for some u, say rt = 1. Then {z, e=,es,...) is a basis for G. A continuous map g : G -----*Zp is determined by the values of g on a basis (provided g(e,~) ~ 0). Therefore by specifying g(en) = f ( e , ) we obtain the desired map. The statement then follows by induction applied to U t x Zp _C_Gp. A Zfextension of k~o/k is given by a continuous homomorhlsm (up to multiplication by an element of Zp) I : G2. - - ~ Zp, This will be a totally ramified (and hence geometric) Zfextension if and only if y(rl u~,) = $~. The splitting of the prime 7~, unramified in kco/k, is determined by the image under ] of the Frobenius automorphism associated to 7~ via class field theory. Fix a geometric Zj,-extension koo/k and let f '. Gt ~
Zp be a continuous homo-
morphism whose kernel fixes koo. Denote by nl the largest integer such that k,tt/k is unramified.
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Gold
THEOREM 1. If k,~ is the n'h-layer
- Kisilevsky
of the Zv-extensJon koo/k then
g,, = o(~,,) > p=("-"')-'/3 and log h,~ _> p2(,,,-,,~)-1 log q/3
for all sufficiently large n. PROOF: By replacing k,~1 by k we may assume that indicated above to compute g,~ we must calculate be the power of ~o dividing Let Z =
koo/k is totally ramified. As w,
D(k,~/k). If P is a prime of k,, let 7~6'
D(k,~/k).
Oal(k,~,~/k7~) and denote by Z = the image under .f of U(v) in Z, wher,
U(v) = 1 + ~r~Fc[[Tr]]. We assume, for simplicity, that 7~ is totally ramified. If not, w must replace Z by the inertia group Z0 in the following argument. Using the Herbran~ function (see [Se]) =
,~ = r
z: Z"le~,
to obtain the lower numbered ramification groups Z~ = Z~(~) = Z ~, we have
,5,, = '~(IZ:l-
1).
Define the integers ro _~ rl _~ r2 pr/~-l. This i
the main difference from the number field case. It follows that k
r
= ,'0
+ ~C,'J - "J-,)pJ j=l
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Gold - K i s i l e v s k y
so that
6. =
~(IZ, I- 1) n
= (1 +
ro)(f'
-
1) +
)--](r
- lb(,k_l))(p
"-k
-
1)
k=l = (1 + r o ) ( f - 1) + ~ ( r ~ k=l > f-~(p
- r~_l)pk(p '~-k - 1)
- 1)~'.-1 + rop'~
>_ ~ o ~ - ' ( p -
~).
We have
g. _> ~ ~.d,gC~,)/2
+ fCgCk)
- 1)
z p~"-=Cp- 13/2- p" 1 2n2= ~v - (v- 1 - 2/f-")
> p2"-2/3 for n sumciently large. Tt follows that [k,, : Fq(z)]/g,, ~ 0 as n ~ oo so we may apply the result of Marian-Madden. Since k,,1 was replaced by k, we see that log h~, ,., 9,, log q > p2('*-"1)-1/3 for n su~ciently large.
REMARK 2 : This result contrasts with the situation in cyclotomic Zfextensions of number s
(see [WaD where
log h~ ~ n p ~ log p or that in constant Zv-exte~ions of function fields over finite fields in which log h,, ,., gp'* log q.
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Gold - K i s i l e v s k y
REMARK 3: Theorem 1 gives only a lower bound r0p 2('~-'~1)-1 logq/3 for logh,~. By choosing Zp-extensions appropriatdy (i.e. by choosing a continuous homomorphism f : Gp ~
Zp such that the sequence of integers {r,~} grows sufficiently rapidly), we get
examples of Zp-extensions, ramified at only one prime, for which the growth rate of log h,~ is arbitrarily large. If we take the example k = Ft(z) then
Gp ~ Zp x I-Iu~,.
Choose any increasing sequence {r,~} of integers such that r . _> pr,,-1, and the continuous homomorphlsm f so that
fCZp) = 0 = f(U~,)
for all
"P # (z)
and on U 1 /(1 - z) = 1
and
.f(1 - x ~) = pk
for p not dividing v and r~-i < v _< rk. For this example
~. >_ (p -- ])prt-lrn_l
w
and
logh. ~ ~=logp.
I w a s a w a theory. In this section we consider the Iwasawa theory for geometric Z~-extensions. As before
let k,, be the n th layer of the geometric Zfextension kc,,/k, and r = G a l ( k ~ / k ) . Let An denote the elements of p-power order in the group T}(k~) of divisor classes of k,~. Let
A~o = limA,~
and
X ~ = limA,,
be the direct and inverse limits of the groups A,~ with respect to the maps induced by inclusion and norm respectively. Let A denote, as usual, the completed group ring Zp[[I']] so that A _~ zp[[Tll. We identify the A-module X ~ with a Galois group as follows:
152
Gold - K i s i l e v s K y PROPOSITION 2. E L m (respectively L,,) is the maximal unramit~ed abelian p-extension
olkm (respecdvely k,,),
then Lm -- UL,,, Xm - Gal(Lm/km) and
C~l( L=/km) ~_ ~ CoI( L./~.) ~- Xoo x Z v.
PROOF: From (I.i)we see that
is a map with dense image and finite kernel of order prime to p (where k = k0 and L = L0). If ~ = Fqk C_ L is the constant (unramifled) Zp-extension of k then
0 --, c,,l(zfi)
-~ Cal(X,/k) ~
Gal('~/k) ---+ 0
is exact. Since A0 is finite and aal(k/k) "-, Zp it follows that p(Ao) c_ aal(L/k). Since the image of D(k) is dense in Gal(L/Ir we see that we must have
c,~ICr,l~) ~ Ao
and
aaICL/~) ~ Ao • z,.
Hence for each ,~, Gal(L,,/k,,) _ A,~ x Zp and since k,,+l = k,,+lk,, it follows that
Gal(L,,+l/'k,~+l) maps onto Gal(L,J'k,~) under restriction so that the norm maps A,,+I surjectively onto A,~. Therefore
_ ~r
•
z,
~- Xoo x Zp.
THEOREM 2. The A-module Xoo has no non-trlvJal finite submodules. H ~herr are ovAy a
s
number of pzlmes (of k) ram//~ed in km/k then Xm is a noetherian torsion A-module.
/ f there are in~z~tely many ramified primes then Xm is not a noetherlan A-module. In this
153
Gold
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Kisilevsky
case TXoo and Xoo/TX= have submodules isomorphic ~o Z ~ and A r has a submodule
isomonohic to (q,/z,,) =. PROOF: We follow Iwasawa [Iw 1]. From El(Gal(k,,/k), F~) = O, it follows that the maps A,, ~
1 with pXi choose p~(z), distinct irreducible polynomials in Fv[x], such that = 1-
This can be done by the analogue of Difieklet's Theorem for Fp(z) (c.f. [Ko].) Then {1 -
z,p~(z)} is a free basis for g(l,) and we define ] : [/(1) ~ f(a
Z, by
- =) = 1
=
o
for,n
i
>
1.
Then for the Zfextension fixed by the kernel of f, the primes corresponding to {p~(z)} are completely decomposed. For the second statement let ~n be the n t~ prime in some ordering and let Zn C Gp be the decomposition group of ~,, in Gp. Then "P,, is totally decomposed in the gp-extension corresponding to the continuous homomorpklsm f : Gp ~
Z~, if and only if J(Z,O = O.
In particular if 0 # z,~ e Z,, then f(z,~) = O. We show that the set of such homomorphisms is gsmall" in the set of all continuous maps. Since Gp is a free abetian prop group a continuous homomorphism is determined by its values on a topological basis. Let B = {e,~} be such a basis (so en -'* O) and let
157
Gold - K i s i l e v s k y B0 = /3 U {0}. Then /30 is compact and the space o[ continuous functions from/30 to Zp is complete with respect to the sup norm. The subspace, C of functions vanishing at 0 is closed and is therefore also complete in the sup norm. Since C is isometric tc Horaco,~z(Gp, Zp) in the sup norm we see that this latter space is also complete. N o w
o,, = { f e ~om=o,.(o~, zp) I f(=,.) = 0} is a closed nowhere dense set in Horaco,u(GT, , Zp), so the set UC,~, of continuous homomorphisms vanishing at some z,~ is of first category (in the sense of Baire category) and hence is a proper subset of Hom=o,,,(Gp, Zp). In fact it is a countable union of "hyperplanes" each of which is of uncountable index (in C) and so is in this sense "small". This argument applies equally well to the Galois group of the maximal abelian pro-p-extension ramified only at some finite set of primes.
Let koo/k be a geometric Zfextension in which only a finite number, r, of primes are ramified and suppose, for simplicity, that they are totany ramified. If
z.(t)
]-[(1 - ~"h) = (1 - ~)(1 - q,)
is the zeta function of k,~, then by Theorem 1 the numerator has degree 29,~ _> ep2'~ for some constant e and for all r~ sufficiently large. Fix a prime 7~ of Q, (the algebraic closure of Q.) over p and let
A,~ = #{a~ n) [ a~'9 = #{unit
is a unit at ~ }
roots of z,,(t)}
and A,, : #{o~") I ^('),*, - 1 mod(~)} = #{1-unit roots of %,~(t)}.
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Gold
Kisilevsky
Then if P l , . . . , 7 9 , are the primes of k,,_l ramified in k,~ it is known that (see e.g. [Ro 2], [Ma] or [D-Ma].) ~,~ - 1 = p(~,.,_, - 1) + (p - 1)deg('P,... 79,) A,~ -
1 -- p(~,.,_,
-
1) -I-
(p -
1)~-.
The term deg(79a ... 79.) is independent of n since the primes 79i are totally ramified. We note that ~,~ is the p-rank of the group of divisor classes of degree 0 in the ~r_ extension kn = ~qk,, obtained by adjoining to k,~ the algebraic closure, Fq, of the constant field, and that 7,, is the p-rank of this group in k,~, so that An is the Iwasawa A-invariant for the Zp-exteusion k,~/k in the usual sense. The recurrence u,, = au~,_l + b
has the solution
~. = ,~(~0 + b / C a - l ) ) -
b/(,-l).
Therefore we have: THEOREM 4. For a Zfextension koo/k as above we have
"~,, = p"(~o + a- 1) + a- 1 ~,, = p " ( L + ~ - - 1) + ~ - - 1 where 0 = d e g ( P l . . . 79.). A / s o
~._
