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Class Number Growth of a Family of Z p-Extensions over Global Function Fields
200, 141]154 Ž1998. JA977233
JOURNAL OF ALGEBRA ARTICLE NO.
Class Number Growth of a Family of Z p-Extensions over Global Function Fields Chaoqun Li and Jianqiang ZhaoU Department of Mathematics, Brown Uni¨ ersity, Pro¨ idence, Rhode Island 02912 Communicated by Walter Feit Received July 31, 1996
1. INTRODUCTION Let Fq be a finite field with q elements and of characteristic p. In this paper, we construct a family of geometric Z p-extensions over global function field k of transcendence degree one over Fq and study the asymptotic behavior of class numbers in such Z p-extensions. By the analog of the Brauer]Siegel theorem in function fields, it suffices to investigate the genus of each layer of such Z p-extensions. In w5x, Gold and Kisilevsky gave a lower bound of the genus for all geometric Z p-extensions of k. They also constructed Z p-extensions whose class numbers grow arbitrarily fast Žsee Remark 3, Section 1 of their paper.. We want to reverse the direction of investigation and try to construct Z p-extensions such that the growth rate is close to the lower bound. It is known that very typical geometric Z p-extensions arise from cyclotomic extensions, both in the number field case and in the function field case. The first systematic study of cyclotomic function fields was carried out in 1930s by L. Carlitz Žsee w1]4x.. Its application in class field theory over rational function fields was found by his student, D. Hayes, in 1974 Žsee w8x.. For generalization to any global function field, we refer the reader to w9x. First we want to fix some notation. Let ` be a fixed prime of k and d` s degŽ`.. Let A be the ring of elements of k holomorphic away from `. We denote by k` the completion of k at ` and V the completion of a fixed algebraic closure of k` . Let F` s Fq d` be the constant field of k` and * Please send correspondence to the second author. E-mail address: [email protected]. edu. 141 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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W` s
2. PRELIMINARIES ON DRINFELD MODULES In this section, we recall some facts on rank one Drinfeld modules and a key proposition which will be crucial for analyzing the higher ramification groups in Section 3. An excellent survey on this topic can be found in Hayes’ paper w11x. One has the following well-known results: ŽH1. w11, Theorem 15.6x Let h be the class number of k. Then the field of constants of HA is F` and w HA : k x s hd` . ŽH2. w11, Proposition 14.1, Ž13.7. and p. 26x The extension Hrk has degree hd`W` rŽ q y 1. and is ramified only at infinite primes. Their decomposition field is HA . Furthermore, the infinite primes are totally and tamely ramified in HrHA . ŽH3. w11, Proposition 4.1x Let B be the integral closure of A in H. Let t : V ª V be the Frobenius q-power map. By definition of H one has r a g Ht 4 where Ht 4 is the non-commutative ring of twisted polynomials i in t so that Ž ar i . ? Ž br j . s ab q t iqj. Let a be an ideal of A. Then the left ideal r a : a g a 4 is generated by a unique monic element ra g Bt 4 . Let
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L a be the set of roots of ra Ž t .. Then L a is a cyclic A-module isomorphic to Ara. ŽH4. w11, p. 28; 10, pp. 226]230x Put H Ž a . s H Ž L a .. Then H Ž a .rk is an abelian extension and the Galois group GalŽ H Ž a .rH . is isomorphic to Ž Ara .=, the group of the invertible elements in Ara. From the similarity with the cyclotomic number fields, H Ž a . is called the a th cyclotomic function field over H. If a s p m is a power of some prime p of A of degree d and P is a prime of H sitting over p, then H Ž p m .rH is totally ramified at P and its ramification degree is F Ž p m . s <Ž Arp m .=< s Ž q d y 1. q dŽ my1.. The only other ramified primes are those over `, with both the inertia group and the decomposition group isomorphic to Fq= . For any a g A and t g V, r aŽ t . has coefficients in B and leading coefficient a unit in B by w9, Corollary 7.4x. We sometimes abbreviate t a for r aŽ t .. Thus the additive polynomial t a can be written as s
ta s
Ý is0
a qi t i
with w ai x g B, w 0a x s a and w as x is a unit in B, where s s deg a. PROPOSITION 2.1. Let P be a prime of B sitting o¨ er a finite prime p of A with deg p s d. Then for any a g A, one has ord P w aj x ) 0 if j - d ? ord p Ž a. and ord P w aj x s 0 if j s d ? ord p Ž a.. Proof. Let ord p Ž a. s l. By ŽH3. of Section 2, there is some b g B such that r aŽ t . s r b Ž rp l Ž t ... Using w9, Proposition 7.6x when l s 1 and using induction for l ) 1 one can show that all but the leading coefficients of rp l Ž t . have positive order at p. Then the proposition follows from the fact that ord P Ž b . s 0. Remark. When k is the rational function field Fq ŽT ., one can actually obtain some equalities in the above proposition using w6, 3.1.11x. See w16, Lemma 1x.
3. HIGHER RAMIFICATION GROUPS In this section, we will compute the ramification groups of H Ž p m .rH. We keep all the previous notations. We further let C be the integral closure of B in H Ž p m . and let G be the Galois group GalŽ H Ž p m .rH .. Then G ( Ž Arp m .= with order F Ž p m .. In what follows we will identify these two groups. Let P˜ be a prime of H Ž p m . lying above P and H Ž p m . P˜
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be the completion of H Ž p m . at P˜. Let H P be the completion of H at P. Then H Ž p m . P˜ is generated over H P by a single root l of f Ž t . s rp m Ž t .rrp my 1 Ž t ., which is Eisenstein at P. So the set li : 0 F i - F Ž p m .4 forms an integral basis for C P˜ over B P . Thus for i G 0 the ith ramification group Gi s a g A : l a ' l Ž mod liq1 . , Ž a, p . s 1 4 . It is clear that G 0 s G because H Ž p m .rH is totally ramified at P. To compute Gi , we need to know l a. THEOREM 3.1. H Ž p m .rH is totally ramified at P˜, i.e., G 0 s G. Moreo¨ er for 0 F i - m y 1 Gq d i s ??? s Gq dŽ iq 1. y 1 ( a g G : ord p Ž a y 1 . G i q 1 4 and Gq dŽ my 1. s 14 . Proof. One only needs to prove the second half of the theorem. Let b s a y 1. We claim that for any q di F j F q dŽ iq1. y 1, l b ' 0 Žmod l jq1 . if and only if ord p Ž b . G i q 1. First we assume i s 0. Then
lb ' 0 «bl ' 0
Ž mod l jq1 . Ž mod l2 . Ž since j ) 1 and r b Ž l . s l b .
«b ' 0
Ž mod l .
«b ' 0
Ž mod p . Ž since b g A . .
Conversely, b'0
Ž mod p .
«lb ' 0
Ž mod l jq1 . Ž since ordl Ž p . G q d y 1 ) j .
« lb ' 0
Ž mod l jq1 . Ž by Proposition 2.1. .
Thus l b ' 0 Žmod l jq1 . if and only if ord p Ž b . G 1 and the case i s 0 of the claim is proved. Suppose i s m y 1, then by ŽH3. of Section 2 one easily sees that ord p Ž b . G m m l b s 0
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which proves the theorem in this case. Now suppose 1 F i F m y 2 and we proceed by induction. For any q di F j F q dŽ iq1. y 1 one has deg
lb s
Ý ts0
b qt l '0 t
Ž mod l jq1 .
m ord p Ž b . G i and
b '0 di
Ž mod l jq1 . Ž by induction .
« ord p Ž b . G i and
b '0 di
Ž mod P . since b g B
ž
di
/
« ord p Ž b . G i q 1. The reason for the last implication is that otherwise ord p Ž b . s i and then ord P w dib x s 0 by Proposition 2.1, which is a contradiction. Conversely, since ordlŽ P . s F Ž p m . ) j if i F m y 2, by Proposition 2.1 ord p Ž b . G i q 1 «
b '0 t
Ž mod P . for 0 F t F d Ž i q 1 . y 1
«
b '0 t
Ž mod l jq1 . for 0 F t F d Ž i q 1 . y 1
deg b
«l s b
Ý ts0
b qt l '0 t
Ž mod l jq1 . .
This proves the claim and the theorem follows at once. 4. GALOIS GROUP Ž Arp m .= In this section, we will study the structure of Ž Arp m .=. The following lemma is a standard result in commutative algebra. LEMMA 4.1. A prŽ p A p . m .
Let A p be the localization of A at p. Then Arp m (
Now we use localization to find some useful facts. LEMMA 4.2. Let R be the localization of A at a prime p of degree d, p its maximal prime ideal, and E its quotient field. Let p be a uniformizer of p.
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Then Ž1. <Ž Rrp m .=< s Ž q d y 1. q dŽ my1.. Ž2. If wi 4 g R, q F i F dr are representati¨ es of a basis of Rrp o¨ er Fp , then = = Ž Rrp m . ( Ž Rrp. = Ł ²1 q wip j :
Ž 1.
i, j
for 1 F i F dr, 1 F j - m and Ž j, p . s 1. Ž3. For 0 F i - m let Bi be the subgroup of Ž Rrp m .= such that Bi s a
½
Ž Rrp m . q
=
s Ž q d y 1 . q dŽ my1.
if i s 0, if 1 F i F m y 1.
dŽ myi.
Proof. Let S s ri g R : 0 F i - q d 4 be the coset representatives of Rrp. Then every element in Ž Rrp m .= has a unique representative in the form a s a0 Ž 1 q a1p q a2 p 2 q ??? qamy1p my 1 . , where a i g S, a0 f p. Parts Ž1. and Ž3. follow from making the choices of a i . For Ž2., let a s 1 q bp j q ??? s 1 q a1p q a2 p 2 q ??? qamy1p my 1 such that b / 0. c Write j s j0 p e, Ž p, j0 . s 1, and choose b 0 g S such that b ' b 0p Žmod p .. This is possible since p e is prime to <Ž Rrp.=< . Thus c
a ' 1 q b 0p p
p e j0
' Ž 1 q bp p j 0 . '
Ž mod p jq1 .
pe
Ž mod p jq1 . e ci p
ž Ł Ž1 q w p . / i
j0
Ž mod p jq1 . ,
where b 0 s Ýc i wi , c i g Z. Hence
a 1 s a Ł Ž 1 q wi p j .
yc i p e
'1
Ž mod p jq1 . .
Repeating this process until j s m one sees that every element in the left hand side of Ž1. lies in the right hand side. Let f i j s min t : jp t G m4 s ulog p Ž mrj .v. Then it is easy to check that p f i j is the exact order of 1 q wip j. Thus by the following lemma, one knows that Ž Rrp m .= and Ž Rrp.== Ł i, j ²1 q wip j : have the same order. Hence the isomorphism in Ž2. is established.
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LEMMA 4.3. my1s
Ý
log p Ž mrj . .
1Fj-m Ž j, p .s1
Proof. Without loss of generality one may assume that p t F m - p tq1 for some nonnegative integer t. If mrp iq1 F j - mrp i then p i - mrj F p iq1 and ulog p Ž mrj .v s i q 1. First, we count the number of j prime to p such that mrp iq1 F j - mrp i. The total number of j such that mrp iq1 F j - mrp i is u mrp i v y u mrp iq1 v. The total number of j such that mrp iq1 F j - mrp i is u mrp i v y u mrp iq1 v. The number of j which is multiple of p so that mrp iq1 - j F mrp i is u mrp iq1 v y u mrp iq2 v. By considering t q 1 such intervals, w mrp tq1, mrp t ., w mrp t , mrp ty1 ., . . . , w mrp, m. one has ist
Ý
log p Ž mrj . s
Ý Ž i q 1. Ž Ž
mrp i y mrp iq1
.
is0
1Fj-m Ž j, p .s1
y Ž mrp iq1 y mrp iq2
...
Let A i s u mrp i v y u mrp iq1 v. Noticing that u mrp tq1 v s 1 and A tq1 s 0 one has t
Ý Ž i q 1. Ž A i y A iq1 . is0 t
s
Ý
t
Ž i q 1. A i y
is0 t
s A0 q
Ý Ž i q 1. A iq1 is0 t
Ý Ž i q 1. A i y Ý iA i is1
is1
t
s A0 q
Ý Ai is1
t
s
ÝŽ
mrp i y mrp iq1
.
is0
s m y 1. Now the proof of the lemma is complete. THEOREM 4.4. Let A be a Dedekind domain, let p be a prime of A of degree d, and p a uniformizer of p. Fix a set of representati¨ es S s wi : 1 F i
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F dr 4 for a basis of Arp o¨ er Fp . Then = = Ž Arp m . ( Ž Arp . =
Ł
1FiFdr 1Fj-m , Ž j, p .s1
²1 q wip j : .
Proof. Use Lemma 4.1 and Lemma 4.2. Remark. Guo and Shu w7, Proposition 3.2x also gave a structure theorem for the group Ž Arp m .= when the base field k is Fq ŽT .. Our result is not as precise as theirs, but on the other hand our proof is much simpler. 5. Z p-EXTENSIONS From the discussion in the previous section, one has =
=
G s Gal Ž H Ž p m . rH . ( Ž Arp m . ( Ž Arp m . =
Ł ²1 q wip j : . i, j
We can assume w 1 s 1 and S s wi : 1 F i F dr 4 . In what follows, we fix a positive number l prime to p Žthe case we are mostly interested in is l s 1.. Let Em be the subfield of H Ž p m . which is the fixed field of the subgroup Ž Arp .== Ł i, j ²1 q wip j :, where the product runs over all 1 q wip j except 1 q p l. Thus GalŽ Em rH . ( ²1 q p l :. This subgroup has order p f m , where f m s ulog p Ž mrl .v. When m gets bigger and bigger, one obtains a Z p-extension of H though generally Em is not the mth layer of the Z p-extension of H. To begin, we compute the genus for each layer of this Z p-extension. We will first compute the difference DE m r H of Em rH. It suffices to know DH Ž p m .r E m and DH Ž p m .r H since DH Ž p m .r H s DH Ž p m .r E m DE m r H . LEMMA 5.1. o¨ er p.
Em is unramified o¨ er H at all primes except for the primes
Proof. H Ž p m . is unramified over H at all primes except for the primes P sitting over p and infinite primes `X . Let L be the fixed field of the subgroup of Ž Arp m .= consisting of all elements of order <Ž Arp .=< . Then the prime `X splits completely in L by ŽH4. of Section 1. Notice that Em is a p-extension of H, `X also splits completely in Em because Em is a subfield of L. LEMMA 5.2. Let N be the Galois group GalŽ H Ž p m .rEm .. Let Gt and Nt be the tth ramification group of H Ž p m .rH and H Ž p m .rEm , respecti¨ ely. Then Ž1. Nt s Gt l N, Ž2. ord P˜Ž DH Ž p m .r H . s Ý`ts0 Ž< Gt < y 1.. Proof. See w15, Chap. II, Propositions 2 and 4x.
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Consider the elements of G s GalŽ H Ž p m .rH . in their direct product forms, i.e., for any a g G, write a as a s aX ŁŽ1 q wip j . a i j , where aX is the non-p part of a. Then, one has the following LEMMA 5.3.
Let a s aX Ž1 q wip j . a i j. Then
a g G1 , G 2 , . . . , Gq dy1 « aX s 1;
a g Gq dŽ my 1. « a s 1.
If 1 F i - m y 1 then a g Gq i d , Gq i dq1 , . . . , Gq Ž iq 1y dy1 « aX s 1, a i j ' 0
Ž mod pwlog
p Ž i r j.xq1
..
Proof. The first statement is trivial to see since all the higher ramification groups in the lemma are p-groups. For 1 F i - m y 1 and q i d F t q Ž iy1. d, if a g Gt then one easily has aX s 1. Now by Theorem 3.1, ord p Ž a y 1. ) i, thus
Ž 1 q wi p j .
ai j
y1'0
Ž mod p iq1 . .
Let e s ord p Ž a i j . then one has jp e ) i which implies that e G wlog p Ž irj .x q1 since e is an integer. We have finished the proof of the lemma. Let p f m be the order of 1 q p l in G. Then f m s ulog p Ž mrl .v. Write e Gi s Ni = ²1 q p l p : for some e. If i s 0 or 1 then e s 0 by Lemma 5.3. Note that Ž l, p . s 1, by Lemma 5.3, if 1 - i - q dŽ my1. then e s wlog p Ž irl .x q 1, thus one has < N0 < s < G 0
< Nq D Ž my 1. < s 1,
< N1 < s ??? s < Nq dy1 < s < G1
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Let P be the prime of Em lying under P˜. Then from Lemma 5.2 ord P Ž DH Ž p m .r E m . s
`
Ý Ž < Ni < y 1. is0
s Ž < N < y 1 . q Ž Ž < N
q
Ý Ž < Gq
di
is1
s 2 < N < y q dŽ my1. my2
q
Ý
q dŽ myiy1. Ž q dŽ iq1. y q di . rp f m ywlog p Ž i r l .xy1
is1
s 2 < N < y q dŽ my1. my2
q
Ý Ž q m d y q dŽ my1. . rp f
m ywlog p Ž i r l .xy1
.
is1
It is elementary to prove that ord P˜ Ž DH Ž p m .r H . s m Ž q m d y q dŽ my1. . y q dŽ my1. , H Ž p m . : Em s < N < s Ž q m d y q dŽ my1. . rp f m . On the other hand ord P˜ Ž DH Ž p m .r H . s ord P˜ Ž DH Ž p m .r E m . q ord P˜ Ž DE m r H . . P is totally ramified one has Noticing that P˜rP ord P Ž DE m r H . s s
ord P˜ Ž DH Ž p m .r H . y ord P˜ Ž DH Ž p m .r E m . H Ž p m . : Em 1
½
m Ž q m d y q dŽ my1. . y q dŽ my1. y 2 < N < q q dŽ my1. my2
y
Ý Ž q m d y q dŽ my1. . rp f is1
my2
s mp f m y 2 y
Ý is1
pwlog p Ž i r l .xq1 .
m ywlog p Ž i r l .xy1
5
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Let f Ž Prp . and g Ž Prp . be the residue class field degree and splitting index for Prp, respectively. Then by Lemma 5.1 deg Ž DE m r H . s
Ý Ý
P
deg P ? ord P Ž DE m r H .
s g Ž Prp . ? deg P ? ord P Ž DE m r H . P . ? ord P Ž DE m r H . s g Ž Prp . ? dim F`Ž CrP P . ? ord P Ž DE m r H . s g Ž Prp . ? dim F`Ž BrP s Ž g Ž Prp . f Ž Prp . rd` . ? d ? ord P Ž DE m r H . s 2 d ? d ? ord P Ž DE m r H . , where d s W` hr2Ž q y 1., since g Ž Prp . f Ž Prp . s w H : k x s 2 d d` . Then by the Hurwitz formula, one has g E m s 1 q 12 deg Ž DE m r H . q Ž g H y 1 . w Em : H x s 1 q d d ord P Ž DE m r H . q Ž g H y 1 . p f m my2
½
s 1 q Ž g H y 1 . p f m q d d mp f m y 2 y
Ý
5
pwlog p Ž i r l .xq1 .
is1
Let L` be the Z p-extension in H Ž L` . s Dm G 1 H Ž p m . corresponding to the subgroup generated by 1 q p l. Take m s l Ž p n y 1.. Then f m s n s ulog p m v. So, L n s Em is the nth layer. Then the genus g n of L n is
½
lp nyly2
1 q Ž g H y 1. p q d d l Ž p y 1. p y 2 y n
n
s l dd p
q dd
ž
Ý
pwlog p Ž i r l .xq1
is1
lp nyly2 2n
n
Ý
5
pwlog p Ž i r l .xq1 y Ž 2 d d y 1 .
is1
/
qŽ gH y 1 y l dd . p n . In order to see the asymptotic behavior of g n , we need the following estimate. p yl y2 wlog p Ž i r l .x LEMMA 5.4. Ýlis1 p ; Ž lrŽ p q 1.. p 2 n as n ª `. n
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Proof. Notice that if i g lp t , lp t q 1, . . . , lp tq1 y 14 , then wlog p Ž irl .x s t. Hence lp nyly2
Ý
pwlog p Ž i r l .x s
is1
ny1
Ý l Ž p tq1 y p t . p i q O Ž p n . ts0
s
lp 2 n pq1
q OŽ pn. .
In the above, we have discussed the Z p-extension corresponding to the group GalŽ E` rH . ( ²1 q p l : which is the topological closure of ²1 q p l :. ŽImplicitly we embedded GalŽ Emy 1rH . into GalŽ Em rH . for all m.. More generally, one could use ²1 q wip j : instead of 1 q p l. One could even use any element a of A such that ord p Ž a y 1. is prime to p. Repeating the argument above, one can get PROPOSITION 5.5. Let L` be the Z p-extension in H Ž L` . s Dm H Ž p m . corresponding to the subgroup generated by 1 q wip l. Then the subfield L n of n n H Ž p lŽ p y1 . fixed by ²1 q wip l : : GalŽ H Ž p lŽ p y1 . .rH . as in Theorem 4.4 is the nth layer of L` and its genus gn ;
1 dd
Ž p q 1.
p2 n s
l dW` 2 Ž p q 1. Ž q y 1.
p2 n
Ž n ª `. .
Now we turn to look at some Z p-extension of k. Assume that k has a prime with its degree relatively prime to p. We regard it as the infinite prime `. We also assume that the class number h of k is relatively prime to p. Let GŽ m. s GalŽ Em rk ., where Em corresponds to one generator 1 q wip l. Then GŽ m. ( GalŽ Em rH . = GalŽ Hrk . is a direct product of p-part with non-p-part, since by ŽH2. of Section 1 the second factor has order W` hd` rŽ q y 1.. Thus there is a unique subgroup of GŽ m. isomorphic to GalŽ Hrk .. Let Fm be the fixed field of this group. THEOREM 5.6. Assume that k has a prime di¨ isor ` whose degree is relati¨ ely prime to the characteristic p of k. Also assume that the class number h of k is relati¨ ely prime to p. Let Fm be constructed as abo¨ e, then Dm Fm is a geometric Z p-extension of k. Let g Xn be the genus of the nth layer FlŽ p ny1. of this extension, and hXn its class number. Then g Xn ; hXn ;
ld 2 Ž p q 1. ld 2 Ž p q 1.
p2 n
Ž n ª `. ,
p 2 n log q
Ž n ª `. .
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153
Proof. By definition Em s H ? Fm , Em is unramified over Fm and Fm is totally ramified over k at p. Let pX be the prime of Fm lying under P. Then deg Ž DF m r k . s d ? ord p X Ž DF m r k . s d ? ord P Ž DE m r H . s deg Ž DE m r H . r2 d . Furthermore by the Hurwitz formula g F m s 1 q 12 deg Ž DF m r k . q Ž g k y 1 . w Fm : k x . From Proposition 5.5, one can easily show the asymptotic for the genus. Next, recall that there is an analog of the Brauer]Seigel theorem due to Madan and Madden Žsee w13x.. It says that if K is a function field with constant field Fq , then log h K ; g K log q as min x g K w K : Fq Ž x .xrg K ª 0, where h K and g K are the class number and the genus of K, respectively. Now one can apply this result in our situation and arrive at the second asymptotic. Remark. If one compares the lower bounds for the genus and class number in w5x Žmore precisely, their proof of the bounds. with the above result, one sees that the growth rates of the genera and class numbers of our family of Z p-extensions are quite close to their bounds. From the penultimate inequality for g n on w5, p. 151x one has lim inf nª`
gn p
2n
G
py1 2 p2
.
If we take l s 1 and assume deg p s 1 in Theorem 5.6 then for our family of Z p-extensions in the theorem lim
nª`
g Xn p
2n
s
1 2 Ž p q 1.
.
It would be interesting to determine if the bound of Gold and Kisilevsky is the best and find some Z p-extension which realizes it. ACKNOWLEDGMENTS Some of the results form part of the first author’s Ph.D. thesis of Brown University Ž1996.. Both authors thank Professor M. Rosen for his encouragement and help. They are also very grateful for the referee’s careful reading of the first draft of this manuscript and herrhis many helpful comments and corrections.
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REFERENCES 1. L. Carlitz, On polynomials in a Galois field, Bull. Amer. Math. Soc. 32 Ž1932., 736]737. 2. L. Carlitz, The arithmetic of polynomials in a Galois field, Amer. J. Math. 54 Ž1932., 39]50. 3. L. Carlitz, On certain functions connected with polynomials in Galois field, Duke Math. J. 1 Ž1935., 137]168. 4. L. Carlitz, A class of polynomials, Trans. Amer. Math. Soc. 43 Ž1938., 167]182. 5. R. Gold and H. Kisilevsky, On geometric Z p-extensions of function fields, Manuscripta Math. 62 Ž1988., 145]161. 6. D. Goss, Fourier series, measures and divided power series in the theory of function fields, K-Theory 2 Ž1989., 533]555. 7. L. Guo and L. Shu, Class numbers of cyclotomic function fields, preprint. 8. D. Hayes, Explicit class field theory for rational function fields, Trans. Amer. Math. Soc. 189 Ž1974., 77]91. 9. D. Hayes, Explicit class field theory in global function fields, in ‘‘Studies in Algebra and Number Theory’’ ŽG. C. Rota, Ed.., pp. 173]217, Academic Press, San Diego, 1979. 10. D. Hayes, Stickelberger elements in function fields, Comp. Math. 55 Ž1985., 209]235. 11. D. Hayes, A brief introduction to Drinfeld modules, in ‘‘The Arithmetic of Function Fields’’ ŽD. Goss, D. Hayes, and M. Rosen, Eds.., de Gruyter, BerlinrNew York, 1992. 12. C. Li ‘‘Z p-Extensions of Function Fields over Finite Fields,’’ Ph.D. thesis, Brown University, 1996. 13. M. L. Madan and D. J. Madden, On the theory of congruence function fields, Comm. Algebra 8 Ž1980., 1687]1697. 14. M. Rosen, The Hilbert class field in function fields, Exposition. Math. 5 Ž1987., 365]378. 15. J.-P. Serre, ‘‘Local Fields,’’ 2nd corrected ed., Graduate Texts in Mathematics, Vol. 67, Springer-Verlag, New York, 1995. 16. J. Zhao, On decomposition law in Carlitz]Kummer function fields, Chinese Sci. Bull. 39, No. 7 Ž1994., 538]540.
JOURNAL OF ALGEBRA ARTICLE NO.
Class Number Growth of a Family of Z p-Extensions over Global Function Fields Chaoqun Li and Jianqiang ZhaoU Department of Mathematics, Brown Uni¨ ersity, Pro¨ idence, Rhode Island 02912 Communicated by Walter Feit Received July 31, 1996
1. INTRODUCTION Let Fq be a finite field with q elements and of characteristic p. In this paper, we construct a family of geometric Z p-extensions over global function field k of transcendence degree one over Fq and study the asymptotic behavior of class numbers in such Z p-extensions. By the analog of the Brauer]Siegel theorem in function fields, it suffices to investigate the genus of each layer of such Z p-extensions. In w5x, Gold and Kisilevsky gave a lower bound of the genus for all geometric Z p-extensions of k. They also constructed Z p-extensions whose class numbers grow arbitrarily fast Žsee Remark 3, Section 1 of their paper.. We want to reverse the direction of investigation and try to construct Z p-extensions such that the growth rate is close to the lower bound. It is known that very typical geometric Z p-extensions arise from cyclotomic extensions, both in the number field case and in the function field case. The first systematic study of cyclotomic function fields was carried out in 1930s by L. Carlitz Žsee w1]4x.. Its application in class field theory over rational function fields was found by his student, D. Hayes, in 1974 Žsee w8x.. For generalization to any global function field, we refer the reader to w9x. First we want to fix some notation. Let ` be a fixed prime of k and d` s degŽ`.. Let A be the ring of elements of k holomorphic away from `. We denote by k` the completion of k at ` and V the completion of a fixed algebraic closure of k` . Let F` s Fq d` be the constant field of k` and * Please send correspondence to the second author. E-mail address: [email protected]. edu. 141 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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W` s
2. PRELIMINARIES ON DRINFELD MODULES In this section, we recall some facts on rank one Drinfeld modules and a key proposition which will be crucial for analyzing the higher ramification groups in Section 3. An excellent survey on this topic can be found in Hayes’ paper w11x. One has the following well-known results: ŽH1. w11, Theorem 15.6x Let h be the class number of k. Then the field of constants of HA is F` and w HA : k x s hd` . ŽH2. w11, Proposition 14.1, Ž13.7. and p. 26x The extension Hrk has degree hd`W` rŽ q y 1. and is ramified only at infinite primes. Their decomposition field is HA . Furthermore, the infinite primes are totally and tamely ramified in HrHA . ŽH3. w11, Proposition 4.1x Let B be the integral closure of A in H. Let t : V ª V be the Frobenius q-power map. By definition of H one has r a g Ht 4 where Ht 4 is the non-commutative ring of twisted polynomials i in t so that Ž ar i . ? Ž br j . s ab q t iqj. Let a be an ideal of A. Then the left ideal r a : a g a 4 is generated by a unique monic element ra g Bt 4 . Let
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143
L a be the set of roots of ra Ž t .. Then L a is a cyclic A-module isomorphic to Ara. ŽH4. w11, p. 28; 10, pp. 226]230x Put H Ž a . s H Ž L a .. Then H Ž a .rk is an abelian extension and the Galois group GalŽ H Ž a .rH . is isomorphic to Ž Ara .=, the group of the invertible elements in Ara. From the similarity with the cyclotomic number fields, H Ž a . is called the a th cyclotomic function field over H. If a s p m is a power of some prime p of A of degree d and P is a prime of H sitting over p, then H Ž p m .rH is totally ramified at P and its ramification degree is F Ž p m . s <Ž Arp m .=< s Ž q d y 1. q dŽ my1.. The only other ramified primes are those over `, with both the inertia group and the decomposition group isomorphic to Fq= . For any a g A and t g V, r aŽ t . has coefficients in B and leading coefficient a unit in B by w9, Corollary 7.4x. We sometimes abbreviate t a for r aŽ t .. Thus the additive polynomial t a can be written as s
ta s
Ý is0
a qi t i
with w ai x g B, w 0a x s a and w as x is a unit in B, where s s deg a. PROPOSITION 2.1. Let P be a prime of B sitting o¨ er a finite prime p of A with deg p s d. Then for any a g A, one has ord P w aj x ) 0 if j - d ? ord p Ž a. and ord P w aj x s 0 if j s d ? ord p Ž a.. Proof. Let ord p Ž a. s l. By ŽH3. of Section 2, there is some b g B such that r aŽ t . s r b Ž rp l Ž t ... Using w9, Proposition 7.6x when l s 1 and using induction for l ) 1 one can show that all but the leading coefficients of rp l Ž t . have positive order at p. Then the proposition follows from the fact that ord P Ž b . s 0. Remark. When k is the rational function field Fq ŽT ., one can actually obtain some equalities in the above proposition using w6, 3.1.11x. See w16, Lemma 1x.
3. HIGHER RAMIFICATION GROUPS In this section, we will compute the ramification groups of H Ž p m .rH. We keep all the previous notations. We further let C be the integral closure of B in H Ž p m . and let G be the Galois group GalŽ H Ž p m .rH .. Then G ( Ž Arp m .= with order F Ž p m .. In what follows we will identify these two groups. Let P˜ be a prime of H Ž p m . lying above P and H Ž p m . P˜
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be the completion of H Ž p m . at P˜. Let H P be the completion of H at P. Then H Ž p m . P˜ is generated over H P by a single root l of f Ž t . s rp m Ž t .rrp my 1 Ž t ., which is Eisenstein at P. So the set li : 0 F i - F Ž p m .4 forms an integral basis for C P˜ over B P . Thus for i G 0 the ith ramification group Gi s a g A : l a ' l Ž mod liq1 . , Ž a, p . s 1 4 . It is clear that G 0 s G because H Ž p m .rH is totally ramified at P. To compute Gi , we need to know l a. THEOREM 3.1. H Ž p m .rH is totally ramified at P˜, i.e., G 0 s G. Moreo¨ er for 0 F i - m y 1 Gq d i s ??? s Gq dŽ iq 1. y 1 ( a g G : ord p Ž a y 1 . G i q 1 4 and Gq dŽ my 1. s 14 . Proof. One only needs to prove the second half of the theorem. Let b s a y 1. We claim that for any q di F j F q dŽ iq1. y 1, l b ' 0 Žmod l jq1 . if and only if ord p Ž b . G i q 1. First we assume i s 0. Then
lb ' 0 «bl ' 0
Ž mod l jq1 . Ž mod l2 . Ž since j ) 1 and r b Ž l . s l b .
«b ' 0
Ž mod l .
«b ' 0
Ž mod p . Ž since b g A . .
Conversely, b'0
Ž mod p .
«lb ' 0
Ž mod l jq1 . Ž since ordl Ž p . G q d y 1 ) j .
« lb ' 0
Ž mod l jq1 . Ž by Proposition 2.1. .
Thus l b ' 0 Žmod l jq1 . if and only if ord p Ž b . G 1 and the case i s 0 of the claim is proved. Suppose i s m y 1, then by ŽH3. of Section 2 one easily sees that ord p Ž b . G m m l b s 0
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which proves the theorem in this case. Now suppose 1 F i F m y 2 and we proceed by induction. For any q di F j F q dŽ iq1. y 1 one has deg
lb s
Ý ts0
b qt l '0 t
Ž mod l jq1 .
m ord p Ž b . G i and
b '0 di
Ž mod l jq1 . Ž by induction .
« ord p Ž b . G i and
b '0 di
Ž mod P . since b g B
ž
di
/
« ord p Ž b . G i q 1. The reason for the last implication is that otherwise ord p Ž b . s i and then ord P w dib x s 0 by Proposition 2.1, which is a contradiction. Conversely, since ordlŽ P . s F Ž p m . ) j if i F m y 2, by Proposition 2.1 ord p Ž b . G i q 1 «
b '0 t
Ž mod P . for 0 F t F d Ž i q 1 . y 1
«
b '0 t
Ž mod l jq1 . for 0 F t F d Ž i q 1 . y 1
deg b
«l s b
Ý ts0
b qt l '0 t
Ž mod l jq1 . .
This proves the claim and the theorem follows at once. 4. GALOIS GROUP Ž Arp m .= In this section, we will study the structure of Ž Arp m .=. The following lemma is a standard result in commutative algebra. LEMMA 4.1. A prŽ p A p . m .
Let A p be the localization of A at p. Then Arp m (
Now we use localization to find some useful facts. LEMMA 4.2. Let R be the localization of A at a prime p of degree d, p its maximal prime ideal, and E its quotient field. Let p be a uniformizer of p.
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Then Ž1. <Ž Rrp m .=< s Ž q d y 1. q dŽ my1.. Ž2. If wi 4 g R, q F i F dr are representati¨ es of a basis of Rrp o¨ er Fp , then = = Ž Rrp m . ( Ž Rrp. = Ł ²1 q wip j :
Ž 1.
i, j
for 1 F i F dr, 1 F j - m and Ž j, p . s 1. Ž3. For 0 F i - m let Bi be the subgroup of Ž Rrp m .= such that Bi s a
½
Ž Rrp m . q
=
s Ž q d y 1 . q dŽ my1.
if i s 0, if 1 F i F m y 1.
dŽ myi.
Proof. Let S s ri g R : 0 F i - q d 4 be the coset representatives of Rrp. Then every element in Ž Rrp m .= has a unique representative in the form a s a0 Ž 1 q a1p q a2 p 2 q ??? qamy1p my 1 . , where a i g S, a0 f p. Parts Ž1. and Ž3. follow from making the choices of a i . For Ž2., let a s 1 q bp j q ??? s 1 q a1p q a2 p 2 q ??? qamy1p my 1 such that b / 0. c Write j s j0 p e, Ž p, j0 . s 1, and choose b 0 g S such that b ' b 0p Žmod p .. This is possible since p e is prime to <Ž Rrp.=< . Thus c
a ' 1 q b 0p p
p e j0
' Ž 1 q bp p j 0 . '
Ž mod p jq1 .
pe
Ž mod p jq1 . e ci p
ž Ł Ž1 q w p . / i
j0
Ž mod p jq1 . ,
where b 0 s Ýc i wi , c i g Z. Hence
a 1 s a Ł Ž 1 q wi p j .
yc i p e
'1
Ž mod p jq1 . .
Repeating this process until j s m one sees that every element in the left hand side of Ž1. lies in the right hand side. Let f i j s min t : jp t G m4 s ulog p Ž mrj .v. Then it is easy to check that p f i j is the exact order of 1 q wip j. Thus by the following lemma, one knows that Ž Rrp m .= and Ž Rrp.== Ł i, j ²1 q wip j : have the same order. Hence the isomorphism in Ž2. is established.
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LEMMA 4.3. my1s
Ý
log p Ž mrj . .
1Fj-m Ž j, p .s1
Proof. Without loss of generality one may assume that p t F m - p tq1 for some nonnegative integer t. If mrp iq1 F j - mrp i then p i - mrj F p iq1 and ulog p Ž mrj .v s i q 1. First, we count the number of j prime to p such that mrp iq1 F j - mrp i. The total number of j such that mrp iq1 F j - mrp i is u mrp i v y u mrp iq1 v. The total number of j such that mrp iq1 F j - mrp i is u mrp i v y u mrp iq1 v. The number of j which is multiple of p so that mrp iq1 - j F mrp i is u mrp iq1 v y u mrp iq2 v. By considering t q 1 such intervals, w mrp tq1, mrp t ., w mrp t , mrp ty1 ., . . . , w mrp, m. one has ist
Ý
log p Ž mrj . s
Ý Ž i q 1. Ž Ž
mrp i y mrp iq1
.
is0
1Fj-m Ž j, p .s1
y Ž mrp iq1 y mrp iq2
...
Let A i s u mrp i v y u mrp iq1 v. Noticing that u mrp tq1 v s 1 and A tq1 s 0 one has t
Ý Ž i q 1. Ž A i y A iq1 . is0 t
s
Ý
t
Ž i q 1. A i y
is0 t
s A0 q
Ý Ž i q 1. A iq1 is0 t
Ý Ž i q 1. A i y Ý iA i is1
is1
t
s A0 q
Ý Ai is1
t
s
ÝŽ
mrp i y mrp iq1
.
is0
s m y 1. Now the proof of the lemma is complete. THEOREM 4.4. Let A be a Dedekind domain, let p be a prime of A of degree d, and p a uniformizer of p. Fix a set of representati¨ es S s wi : 1 F i
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LI AND ZHAO
F dr 4 for a basis of Arp o¨ er Fp . Then = = Ž Arp m . ( Ž Arp . =
Ł
1FiFdr 1Fj-m , Ž j, p .s1
²1 q wip j : .
Proof. Use Lemma 4.1 and Lemma 4.2. Remark. Guo and Shu w7, Proposition 3.2x also gave a structure theorem for the group Ž Arp m .= when the base field k is Fq ŽT .. Our result is not as precise as theirs, but on the other hand our proof is much simpler. 5. Z p-EXTENSIONS From the discussion in the previous section, one has =
=
G s Gal Ž H Ž p m . rH . ( Ž Arp m . ( Ž Arp m . =
Ł ²1 q wip j : . i, j
We can assume w 1 s 1 and S s wi : 1 F i F dr 4 . In what follows, we fix a positive number l prime to p Žthe case we are mostly interested in is l s 1.. Let Em be the subfield of H Ž p m . which is the fixed field of the subgroup Ž Arp .== Ł i, j ²1 q wip j :, where the product runs over all 1 q wip j except 1 q p l. Thus GalŽ Em rH . ( ²1 q p l :. This subgroup has order p f m , where f m s ulog p Ž mrl .v. When m gets bigger and bigger, one obtains a Z p-extension of H though generally Em is not the mth layer of the Z p-extension of H. To begin, we compute the genus for each layer of this Z p-extension. We will first compute the difference DE m r H of Em rH. It suffices to know DH Ž p m .r E m and DH Ž p m .r H since DH Ž p m .r H s DH Ž p m .r E m DE m r H . LEMMA 5.1. o¨ er p.
Em is unramified o¨ er H at all primes except for the primes
Proof. H Ž p m . is unramified over H at all primes except for the primes P sitting over p and infinite primes `X . Let L be the fixed field of the subgroup of Ž Arp m .= consisting of all elements of order <Ž Arp .=< . Then the prime `X splits completely in L by ŽH4. of Section 1. Notice that Em is a p-extension of H, `X also splits completely in Em because Em is a subfield of L. LEMMA 5.2. Let N be the Galois group GalŽ H Ž p m .rEm .. Let Gt and Nt be the tth ramification group of H Ž p m .rH and H Ž p m .rEm , respecti¨ ely. Then Ž1. Nt s Gt l N, Ž2. ord P˜Ž DH Ž p m .r H . s Ý`ts0 Ž< Gt < y 1.. Proof. See w15, Chap. II, Propositions 2 and 4x.
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Consider the elements of G s GalŽ H Ž p m .rH . in their direct product forms, i.e., for any a g G, write a as a s aX ŁŽ1 q wip j . a i j , where aX is the non-p part of a. Then, one has the following LEMMA 5.3.
Let a s aX Ž1 q wip j . a i j. Then
a g G1 , G 2 , . . . , Gq dy1 « aX s 1;
a g Gq dŽ my 1. « a s 1.
If 1 F i - m y 1 then a g Gq i d , Gq i dq1 , . . . , Gq Ž iq 1y dy1 « aX s 1, a i j ' 0
Ž mod pwlog
p Ž i r j.xq1
..
Proof. The first statement is trivial to see since all the higher ramification groups in the lemma are p-groups. For 1 F i - m y 1 and q i d F t q Ž iy1. d, if a g Gt then one easily has aX s 1. Now by Theorem 3.1, ord p Ž a y 1. ) i, thus
Ž 1 q wi p j .
ai j
y1'0
Ž mod p iq1 . .
Let e s ord p Ž a i j . then one has jp e ) i which implies that e G wlog p Ž irj .x q1 since e is an integer. We have finished the proof of the lemma. Let p f m be the order of 1 q p l in G. Then f m s ulog p Ž mrl .v. Write e Gi s Ni = ²1 q p l p : for some e. If i s 0 or 1 then e s 0 by Lemma 5.3. Note that Ž l, p . s 1, by Lemma 5.3, if 1 - i - q dŽ my1. then e s wlog p Ž irl .x q 1, thus one has < N0 < s < G 0
< Nq D Ž my 1. < s 1,
< N1 < s ??? s < Nq dy1 < s < G1
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Let P be the prime of Em lying under P˜. Then from Lemma 5.2 ord P Ž DH Ž p m .r E m . s
`
Ý Ž < Ni < y 1. is0
s Ž < N < y 1 . q Ž Ž < N
q
Ý Ž < Gq
di
is1
s 2 < N < y q dŽ my1. my2
q
Ý
q dŽ myiy1. Ž q dŽ iq1. y q di . rp f m ywlog p Ž i r l .xy1
is1
s 2 < N < y q dŽ my1. my2
q
Ý Ž q m d y q dŽ my1. . rp f
m ywlog p Ž i r l .xy1
.
is1
It is elementary to prove that ord P˜ Ž DH Ž p m .r H . s m Ž q m d y q dŽ my1. . y q dŽ my1. , H Ž p m . : Em s < N < s Ž q m d y q dŽ my1. . rp f m . On the other hand ord P˜ Ž DH Ž p m .r H . s ord P˜ Ž DH Ž p m .r E m . q ord P˜ Ž DE m r H . . P is totally ramified one has Noticing that P˜rP ord P Ž DE m r H . s s
ord P˜ Ž DH Ž p m .r H . y ord P˜ Ž DH Ž p m .r E m . H Ž p m . : Em 1
½
m Ž q m d y q dŽ my1. . y q dŽ my1. y 2 < N < q q dŽ my1. my2
y
Ý Ž q m d y q dŽ my1. . rp f is1
my2
s mp f m y 2 y
Ý is1
pwlog p Ž i r l .xq1 .
m ywlog p Ž i r l .xy1
5
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OVER FUNCTION FIELDS
Let f Ž Prp . and g Ž Prp . be the residue class field degree and splitting index for Prp, respectively. Then by Lemma 5.1 deg Ž DE m r H . s
Ý Ý
P
deg P ? ord P Ž DE m r H .
s g Ž Prp . ? deg P ? ord P Ž DE m r H . P . ? ord P Ž DE m r H . s g Ž Prp . ? dim F`Ž CrP P . ? ord P Ž DE m r H . s g Ž Prp . ? dim F`Ž BrP s Ž g Ž Prp . f Ž Prp . rd` . ? d ? ord P Ž DE m r H . s 2 d ? d ? ord P Ž DE m r H . , where d s W` hr2Ž q y 1., since g Ž Prp . f Ž Prp . s w H : k x s 2 d d` . Then by the Hurwitz formula, one has g E m s 1 q 12 deg Ž DE m r H . q Ž g H y 1 . w Em : H x s 1 q d d ord P Ž DE m r H . q Ž g H y 1 . p f m my2
½
s 1 q Ž g H y 1 . p f m q d d mp f m y 2 y
Ý
5
pwlog p Ž i r l .xq1 .
is1
Let L` be the Z p-extension in H Ž L` . s Dm G 1 H Ž p m . corresponding to the subgroup generated by 1 q p l. Take m s l Ž p n y 1.. Then f m s n s ulog p m v. So, L n s Em is the nth layer. Then the genus g n of L n is
½
lp nyly2
1 q Ž g H y 1. p q d d l Ž p y 1. p y 2 y n
n
s l dd p
q dd
ž
Ý
pwlog p Ž i r l .xq1
is1
lp nyly2 2n
n
Ý
5
pwlog p Ž i r l .xq1 y Ž 2 d d y 1 .
is1
/
qŽ gH y 1 y l dd . p n . In order to see the asymptotic behavior of g n , we need the following estimate. p yl y2 wlog p Ž i r l .x LEMMA 5.4. Ýlis1 p ; Ž lrŽ p q 1.. p 2 n as n ª `. n
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Proof. Notice that if i g lp t , lp t q 1, . . . , lp tq1 y 14 , then wlog p Ž irl .x s t. Hence lp nyly2
Ý
pwlog p Ž i r l .x s
is1
ny1
Ý l Ž p tq1 y p t . p i q O Ž p n . ts0
s
lp 2 n pq1
q OŽ pn. .
In the above, we have discussed the Z p-extension corresponding to the group GalŽ E` rH . ( ²1 q p l : which is the topological closure of ²1 q p l :. ŽImplicitly we embedded GalŽ Emy 1rH . into GalŽ Em rH . for all m.. More generally, one could use ²1 q wip j : instead of 1 q p l. One could even use any element a of A such that ord p Ž a y 1. is prime to p. Repeating the argument above, one can get PROPOSITION 5.5. Let L` be the Z p-extension in H Ž L` . s Dm H Ž p m . corresponding to the subgroup generated by 1 q wip l. Then the subfield L n of n n H Ž p lŽ p y1 . fixed by ²1 q wip l : : GalŽ H Ž p lŽ p y1 . .rH . as in Theorem 4.4 is the nth layer of L` and its genus gn ;
1 dd
Ž p q 1.
p2 n s
l dW` 2 Ž p q 1. Ž q y 1.
p2 n
Ž n ª `. .
Now we turn to look at some Z p-extension of k. Assume that k has a prime with its degree relatively prime to p. We regard it as the infinite prime `. We also assume that the class number h of k is relatively prime to p. Let GŽ m. s GalŽ Em rk ., where Em corresponds to one generator 1 q wip l. Then GŽ m. ( GalŽ Em rH . = GalŽ Hrk . is a direct product of p-part with non-p-part, since by ŽH2. of Section 1 the second factor has order W` hd` rŽ q y 1.. Thus there is a unique subgroup of GŽ m. isomorphic to GalŽ Hrk .. Let Fm be the fixed field of this group. THEOREM 5.6. Assume that k has a prime di¨ isor ` whose degree is relati¨ ely prime to the characteristic p of k. Also assume that the class number h of k is relati¨ ely prime to p. Let Fm be constructed as abo¨ e, then Dm Fm is a geometric Z p-extension of k. Let g Xn be the genus of the nth layer FlŽ p ny1. of this extension, and hXn its class number. Then g Xn ; hXn ;
ld 2 Ž p q 1. ld 2 Ž p q 1.
p2 n
Ž n ª `. ,
p 2 n log q
Ž n ª `. .
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153
Proof. By definition Em s H ? Fm , Em is unramified over Fm and Fm is totally ramified over k at p. Let pX be the prime of Fm lying under P. Then deg Ž DF m r k . s d ? ord p X Ž DF m r k . s d ? ord P Ž DE m r H . s deg Ž DE m r H . r2 d . Furthermore by the Hurwitz formula g F m s 1 q 12 deg Ž DF m r k . q Ž g k y 1 . w Fm : k x . From Proposition 5.5, one can easily show the asymptotic for the genus. Next, recall that there is an analog of the Brauer]Seigel theorem due to Madan and Madden Žsee w13x.. It says that if K is a function field with constant field Fq , then log h K ; g K log q as min x g K w K : Fq Ž x .xrg K ª 0, where h K and g K are the class number and the genus of K, respectively. Now one can apply this result in our situation and arrive at the second asymptotic. Remark. If one compares the lower bounds for the genus and class number in w5x Žmore precisely, their proof of the bounds. with the above result, one sees that the growth rates of the genera and class numbers of our family of Z p-extensions are quite close to their bounds. From the penultimate inequality for g n on w5, p. 151x one has lim inf nª`
gn p
2n
G
py1 2 p2
.
If we take l s 1 and assume deg p s 1 in Theorem 5.6 then for our family of Z p-extensions in the theorem lim
nª`
g Xn p
2n
s
1 2 Ž p q 1.
.
It would be interesting to determine if the bound of Gold and Kisilevsky is the best and find some Z p-extension which realizes it. ACKNOWLEDGMENTS Some of the results form part of the first author’s Ph.D. thesis of Brown University Ž1996.. Both authors thank Professor M. Rosen for his encouragement and help. They are also very grateful for the referee’s careful reading of the first draft of this manuscript and herrhis many helpful comments and corrections.
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