a-= E K · 2018. 11. 16. · where \pk = \pu~k and Bk(^k) stands for the Arth generalized Bernoulli number attached to the character \pk. Denote by Q^x) the extension of Qp generated

MATHEMATICS OF COMPUTATIONVOLUME 49, NUMBER 179JULY 1987, PAGES 281-294

A Method for Computing the Iwasawa \-Invariant

By R. Ernvall and T. Metsänkyla

Abstract. We present a method for computing the minus-part of the Iwasawa X-invariant ofan Abelian field K. Applying this method, we have computed X' for several odd primes pwhen K runs through a large number of quadratic extensions of the pú\ cyclotomic field. Areport on these computations and an analysis of the results is included.

1. Introduction. Let K be an Abelian field, i.e., a finite Abelian extension of Q.For a prime p > 2, consider the cyclotomic Z^-extension Kx of K. Let K„ (n > 0)denote the intermediate field of Kx/K which is cyclic of degree p" over K. The/7-part of the class number of K„ equals px"+p, for all sufficiently large n, whereA = X(p) and v = v(p) are integral constants, A > 0. Call A the Iwasawa A-invariant of K and write A = A++ A", where A+ is the corresponding invariant ofthe maximal real sub field of K. In this paper we present a method for computingA ", developed by the second author, and report on computer calculations by the firstauthor, performed by this method.

If the conductor fK of the field K is divisible by p2, then K has a subfield L suchthat p2 + fL and the cyclotomic Z^-extension of L equals Kx. Hence we assume,without loss of generality, that p21 fK. Denote by C\\(K) the character group of K.It is known that A" decomposes as

a-= E K

with

X = X(K) = {x e Ch(K): x(-l) --1, X * «J'1},

where u denotes the Teichmüller character mod/7 and Ax is the A-invariant of theIwasawa power series representing the /?-adic L-function Lp(s, x«).

Thus, the computation of A~ is reduced to the determination of the componentsA . This will be done in two steps: We first relate Ax to the /^-orders of certaingeneralized Bernoulli numbers and then show how to determine these ^-orders bymeans of a series of character sum congruences. As an application we consider thefields Q(Jm,Çp), where m is an integer prime to p and $p denotes a primitive /Hhroot of 1. In this case the congruences in question are simply rational congruencesmod p.

Received September 5, 1986.1980 Mathematics Subject Classification (1985 Revision). Primary 11R23, UR29,11S40,11Y40, 11-04.

'»1987 American Mathematical Society0025-5718/87 $1.00 + $.25 per page

281

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

282 R. ERNVALL AND T. METSÁNKYLA

The computational part of our work consists of the determination of A~ for quitea large collection of fields Q(Jm,Çp), chosen so that either p or \m\ is small. Moreprecisely, we computed for these fields the components Ax with x = 6mu', where 6mis the quadratic character of the field Q(Jm); this is sufficient since the A~-invariants of the cyclotomic subfields Q(Çp) are known. Our results also give theA-invariant of Q(vW) for the negative m in the range under consideration.

There are previous numerical results about A" in [1], [2], [3], [4], [6]. These concernmainly quadratic fields and the fields Q(J-ï,Çp) and Q(v/-3~,fi,), and in all casesthe decomposition of A~ is simple in the sense that either there is but one positivecomponent Ax, or all the positive components are equal to 1. In the present resultsthis is no longer the case.

A detailed description of our computations appears in Sections 7-9.

2. On p-Ad\c L-Functions. For the theory of this section, the reader is referred toWashington's book [11], in particular to Sections 5.2 and 7.2.

We fix an embedding of the field of algebraic numbers in an algebraic closure Qpof Q , the field of p-adic numbers. Denote by ord^, the p-adic valuation on üp,normalized so that ordp(p) = I.

Let x be a character in X(K) (all characters are assumed primitive). Since theconductor / of x divides fK, it is not divisible by p2; we say that x is of the "firstkind". Put

/ = d or dp with ( d, p ) = 1.

As in the introduction, let Kn denote the «th layer of the Z^-extension Kx/K(n > 0). The character group of Kn is of the form Ch(AT) X («■„), where irn is acharacter with order pn and conductor p"+1 (or 1, if n = 0); irn is called a characterof the "second kind".

Now consider the /?-adic L-function Lp(s,\p) for the (nonprincipal) character1),

where \pk = \pu~k and Bk(^k) stands for the Arth generalized Bernoulli numberattached to the character \pk.

Denote by Q^x) the extension of Qp generated by the values of x- Iwasawa'stheory of />adic L-functions asserts that there exists a power series

00

(2) /Ux«)=E¥y7-0

whose coefficients ay = ay(x) are integers of Qp(x), such that

According to the Ferrero-Washington theorem, the power series f(x, x

A METHOD FOR COMPUTING THE IWASAWA X-INVARIANT 283

3. The /»-Orders of Generalized Bernoulli Numbers. Let us decompose x as

X = du'1 with fg = d (^ I), l

284 R. ERNVALL AND T. METSÄNKYLÄ

Proof. Suppose that the above congruence holds. If Ax < (p")/e, then acomparison of this congruence with the first part of Proposition 1 shows thatXx^ h. If Ax ^ (/»")/e, then the assertion follows directly from the assumptionmade about h.

To verify the converse, apply both parts of Proposition 1 separately. DRemark. Proposition 2 is of the same kind as the main result in the second

author's paper [8]. This relates Ax to certain Kummer type congruences of Bk(0),provided Ax ^ p - 1. Proposition 2 would enable one to replace the proof presentedin [8] by a somewhat simpler proof.

4. Bernoulli Numbers and Character Sums. We now express the residue of B'(6irn)modulo p in terms of suitable character sums.

For any character t/> with conductor / we have, in the usual symbolic notation,

Bk(t) = \Zt(a)(fB + a-f)k (k > 0)

(e.g., [7, p. 134]), where the Bm denote ordinary Bernoulli numbers. On changing thesummation variable a into / - a we obtain

BkW={-1)kt{-1]ÍUa)(a-fB)kf a=\

= A,.,»-!Let \p = 6irn with n > I. Then \p(-l) = (-1)' since the character x = Ouodd and irn, being of /»-power order, is even. Hence we find that

is

B'(6irn)1 dp"

dpH+l(6Trn)(a)(a - dp" + lB)'

a = l

-7^71 I (^J(a)a'-^1 I (i?77j(a)a'-1(mod/»).dp-

dp" a-\ a-l

The second sum of the last expression vanishes mod/», as can be verified again bythe transformation a -» dp"+l — a. Therefore,

dp"*1

(4)1B'(«Os^TT S {6irn)(a)a'(modp).

u = l

From this result we derive the following congruence which is of the same type as theclassical Voronoï congruence for ordinary Bernoulli numbers. We point out that thecongruence (in a sharper form) has also been proved by Slavutskil [9, congr. (6)].

Proposition 3. Let b be a positive rational integer with (b, dp) = 1. Then

(b 1 and 1 < t < /» — 1.

Proof. Put \p = 6irn. Let a and b be positive rational integers prime to dp.Keeping b fixed, we write

baba = dp"dpn+\

+ ra, 0


On raising this equation to the rth power and multiplying by ty(a) = \p(b) ty(/"a),we get

ba^(a)b,al = *{b)-\(ra)r: + l(a)tr^dp/-ij„« + i

dpn + i(mod/»2" + 2).

ba

If a runs through 1,..., dpn + l, excepting those numbers for which (a, dp) > 1, thenso does ra. Summing over a we find that (observe that ^(a) = 0 if (a, dp) > I)

(¿' - t(b)-1) E 4,(a)a' - tdp"^ E ^(a).r1a=l a=l

Since ra = ¿»a (mod/»" + 1), this result together with (4) yields the claimed con-gruence. D

dpn + 1(mod/»2"+2).

5. The Main Result. Every rational integer a prime to p has the following uniquerepresentation mod p"+1:

(5) a = "-!),

where the sum is extended over those numbers a for which 1 < a < a/»"+1, (a, a/»)= 1 and v(a) = k. Moreover, set

(7) Tu=Tr=Pl!(ku)snk (u = 0,...,p"-l).k = u

Theorem. Let x = 6u'~l e X(K), where fe = d is prime top and 1 < t h ifandonlyif E 0(a)ir„(a)a'-1a=l

where k = l/(/»").For a fixed n > 1, write

t„(1 +p) = 1 +T).

Then we have ordp(Tj) = k and, by (5),

77„(a) = (l+rJ)"


Consequently,dp-*'

E 0(a)irn(a)a


Now let us enlarge the field to K = Q(Jm , Çp) with p \ m. Then the character setX is enlarged by the characters 6mu'x discussed in Section 6. To be precise, we have

A-=A0 +E*X,

(10) X = du- with

where the sum is extended over the characterst = 2,4,...,/? - 1 ifw>0,t = 1,3,...,/» - 2 ifw 6 (see below).

The numerical material thus obtained contains about 22000 values of Ax, some6400 of them being positive. Samples from this material are exhibited in Tables 1and 2 of the appendix. Table 1 presents the results for p = 5, m > 0, and Table 2for p < 200, m = -1, ±2, -3, 5, -7. Note that every odd prime /» below 200 reallyappears in Table 2, i.e., to every p there is at least one m and t such that Ax > 0 forX = 6mu-\

For /» > 3, only few cases were found in which Ax > p - 2. These cases, whichhad to be settled on the level n = 2, are listed here:

4391427-311-761-966

-2861-3081

-3178-3471-3547-3923-4026-4774-1371

For p = 7 it in fact turned out that Ax varies between 0 and 4 (assuming all values0,..., 4) except in the single case given above. For /» = 11 we have the maximumAx = 3 form = -723, r= 1.

If p = 3, then A>l(=/»-2)in about a third of the cases. These could besettled on the level n = 2 (i.e., Ax < 5), except in six cases. In the latter cases thecontinuation of the procedure was given up since the values of A can be found in[6]; they are as follows:

Ax = 6 for m = -239, -1022, -1427, -1777;for m =

for m =

-458,-2789.



An examination of the results shows that the values of Ax seem to be distributedin the expected way. For example, if we keep /» and / fixed, t ¥= 1, and let m vary,then the number of cases with Ax > k (for x = 0mu'~x and k > 0) should be abouta /»*th part of the number of all Ax; this corresponds to the natural hypothesis thatthe coefficients of the power series f(x,x k in our range:

%157715961596253515991599

A/,

553326329490221256

N,172556S882939

509

1514

NJK0.350.200.210.190.140.16

X

l/p0.330.200.200.200.140.14

Ni/No0.110.0340.0430.0350.0180.024

W0.110.0400.0400.0400.0200.020

N3/N0

0.0320.0060.0090.006

W0.0370.0080.0080.008

If t = 1, the situation is different. Indeed, by Eqs. (l)-(3) the constant term off(x, xw) equals(11) a0=(x(p)-l)Bx(x);hence, in the present case Ax is positive whenever x(p) = 0m(p) = +1. We musttherefore modify the above hypothesis so as to concern those f(x,0mu) only forwhich 6m(p) = -1. We tested this hypothesis for p = 5, m > 0, obtaining thefollowing (Nk denotes the number of Ax ^ k when 0m(5) = -1):

N¿ = 1268, AY = 241, 7V2'= 36; Nx'/N¿ = 0.19, N{/N¿ = 0.028.We may also ask how often A~ is, say, positive as p is fixed and \m\ increases. If

/» < 11, then Xq = 0, and so A"> 0 exactly when at least one of the s = (p - l)/2numbers T0 corresponding to the characters 6mu'~l vanishes mod p. To avoid theexceptional case t = I, consider positive m only. Then it is again natural to assumethat the values of T0 be randomly distributed mod /», and this implies that theproportion of the number of fields with A~> 0 to the number of all fields should beabout p = 1 — (1 — p'l)s. Below is a comparison between the observed and ex-pected values of this proportion:

observed proportion

57

11

587/1596 = 0.37204/530 = 0.38100/279 = 0.36

0.360.370.38

A table including all the results of our computations has been deposited in theUMT file; see Review 29 in this issue.

8. Comparison with Previous Results. We next describe the contents of thepreviously published tables about A". These tables were used by us to check ourresults.

Gold [3], [4] has computed, for /» = 3, 5, 7, 11, the A-invariant of the quadraticfield with discriminant -d < 0. His results in [4, Table 2] cover the range 0 < d < 264.They agree completely with ours, and so do also the additional results presented in[3, Tables 2 and 5] after the following apparent errors are corrected: In Table 2, thevalue 1253 for d should be 1263 (corresponding to the given class number 20); in



Table 5, lines 5 and 6, instead of A = 3 and A = 4 one should read A = 2. The lattercorrection is confirmed not only by [6] quoted below, but also by Corollary 5 in [3].The expressions for en in Table 5 should be correspondingly corrected.

Kobayashi [6] investigates, for p = 3, the power series f(x, xu) with x = #„, andX = 6mu. He has determined the coefficients a0,_a8 mod 9 of this power seriesfor -104 < m < 0 and 0 < 3m < 104. From his table one can read the value of Ax,since in all cases Ax < 8. Note that for x = 6m the table is far more extensive thanours, while for x = 0mu our computations go a bit farther. The overlapping parts ofboth tables are in agreement, except that the table in [6] omits the first negative mwith Ax > 0, namely m = -2. The nonvanishing of Ax in this case follows, by (11),from the fact that x(3) = 0_2(3) = +1. Our computation indeed shows, in agree-ment with [4], that Ax = 1.

The first author has determined, for /» < 104, the components A with x = 9mu'~lfor m = -1 and m = -3 (see [2] and [1], respectively). For t = 3,5,...,/»- 2, onehas in this range A = 1 if (/>, r — 1) is an ¿-irregular or D-irregular pair, respec-tively, and Ax = 0 otherwise. A comparison of the tables in [1] and [2] with thepresent Table 2 shows no discrepancies.

The paper [5] by Hao and Parry tabulates the "w-irregular" primes /» < 5025 forthe values of m that appear in our Table 2. For a fixed m, the prime /» is w-irregularif and only if there is at least one t > 1 such that Ax > 0 with x = 0mu'~l. It iseasily checked that, for /» < 200, the lists given in [5] coincide with the corre-sponding lists extracted from Table 2. Our computations show the somewhatinteresting fact that every positive value of Ax in this region in fact equals 1, exceptfor a single value Ax = 2 occurring for p = 23 and x = 0_2uw.

Let us finally mention that if m = -q, with q a prime, and 0m(p) = -1, then itfollows from (11) that XB > 0 exactly when the class number of the field Q(\f-q) isdivisible by p. Thus a partial check of our results is also provided by the classnumber tables of imaginary quadratic fields.

9. The computations. The computations were run on the DEC-20 computer at theUniversity of Turku. The programs, written in Fortran, used only integer arithmetic.

As is seen from Sections 5 and 6, the main task was the computing of the sums Snk(mostly for n = 1). This was started by searching a primitive root mod p andconstructing the index table. After decomposing m into prime factors, the charactervalues 0m(a) were calculated via the Legendre symbol, using the congruence

I — ) = aíq~l)/2 (moda) (a an odd prime factor of m)

and then checking that 6m(a) indeed equals +1 or 0. For a fixed t, we chose aminimal b > 0 such that (b,dp) = 1 and 6m(b)b' ^ l (mod /»). To find the value ofv(a) for n = 1 (see (6) and (5)), we computed a^1 mod p2 by employing the 2-adicexpansion of /» - 1 and the residues of a2, a4, aK,... mod p2.

After computing the numbers SXkmodp we searched for the first nonvanishingnumber in the sequence ro(1),..., TpiX}2 mod p. The cases in which such a number didnot exist were afterwards picked out by hand and dealt with on the level n = 2. Theprocedure on this level was similar, except that this time the determination of v(a)required computations mod /»3.



Appendix

The positive values of Ax for p

Table 15, x = 0mu'~l (t = 2 or 4) and 0 < m ^ 3147.

m t X m t X

27127828128228729329829830731331 4326347358366382391398401407

426426

43343443843944645345745 7458466467467469471473479489497498499499501501502

509509514519

534537541543554559

577581581587587589591597602606611617

62362662662762962963163363464365 4662673674678679681683687689699717719

74174375375375475875976176376676 7781789791

809814817822839842651857861869874881881887889893903911917921922922923926933937939943946947949


A METHOD FOR COMPUTING THE IWASAWA X-INVARIANT

Table 1 (continued)

291

m t X m A m XJL

m

973978982983986997

10031006100710181031103410371041104210511059106310691073107410771079108610871093109711061111111311131114111811191121112211231126112611291131113311371137114211 4911571169117311 82118711 91119311 94119812031213

12141217

123112311238123812431247125312541261126212671273127912791281128912911293129413011313131713211327132713381339134213511354136613661379138213891389139313981401140214061407142614271429143414341441144314511461146614781479

1483I4861493149315061509151115141518151815291531153115331541154615711577157915821586159715971621163116311633163716411654165816621663168616991702171317171721172317311738173817391741175417571758175817611762176617691777177917831766

■I 79717981 79913031817182918291834183418371838184618471851185318531861187418821891189718981907190719131913191 419141921192319341937193819381941194319491954195 71959196619691973197719791982198619991 99920012002200220032:003201 420272027


292 R. ERNVALL AND T. METSÄNK.YLÄ

Table 1 (continued)

m m m m

20292031203320382039¿0512081208320832087208920932098210121 0221112119212221 23212621 2721292131214 321 462153215321572158215921 6121712177218 3218221892189219422192221

22292231223422432243225 7:L. ¿_ O ..'...

226322632271227322782279226122872306

230723072314231723232326

233323342341234223472353

2381238623862391239123972399240624112433243S24382446244924592462247124812482248624872489249825012503250325092513251925222526253125332534254625512557255825632566

25712573257325 77257825 792581259926022603260926332634263926472654

■.'-..'

266 I2669267126712683268626872667269426982706271127142 7 2 22723272327312739274127422 743274627592766277127732 776280328142821.2822282°2829284126432851285928772878

287828822886288729112914.29232927293129462949295 929632966296729712973297429632986299129912993299429983013301430233039304130533054305930773082309 831013102310331063107311131133121312131263127312931293134313831 47



Table 2

The values oft, 1 ^ t < p - 1, for which Xx > 0 with x = 0mw'-1, '" the

region 2 < p < 200, m = -7, -3, -2, -1, 2, 5. 77ie dagger (') indicatesthat Xx = 2; in all other cases Ax = 1.

357

1 1

13

17

19

2329

31

3741

43475 3

59

61

67

7173

79

83

89

97

-3 -2

119

125

119433 3

71 1

5365

1

1 7

11

1

1 3294 5

147

1

11 1

1 1

119

131

115

13 3

2 311

1 3

15

1

17

27

291

19

412

3034

343650

68

1 630

32

1 4

18

42

6

70

±1101

103

107

109

113

127

1 31

137

139

149

1511 57

163167

1 73

179

181

191

193

197

199

1105

165

145

10121

139

1031

101

1

1 397

1531

35177

131

159

193

55

199

113

109

15157

157

119

79

121

11 19

179183

1161

191

163

143

129

1147

175

6486

405 6

146

74

74

22100

62

3264

84

44104

22

66

1446 b

10

186



Department of MathematicsUniversity of TurkuSF-20500 Turku, Finland

1. R. Ernvall, "Generalized Bernoulli numbers, generalized irregular primes, and class number,"Ann. Univ. Turku. Ser. A I, No. 178,1979, 72 pp.

2. R. Ernvall & T. Metsänkylä, "Cyclotomic invariants and £-irregular primes," Math. Comp., v.32,1978, pp. 617-629.

3. R. Gold, "Examples of Iwasawa invariants," Acta Arith. v. 26,1974, pp. 21-32.4. R. Gold, "Examples of Iwasawa invariants, II," Actu A nth., v. 26,1975, pp. 232-240.5. F. H. Hao & C. J. Parry, "Generalized Bernoulli numbers and »¡-regular primes," Math. Comp.,

v. 43, 1984, pp. 273-288.6. S. Kobayashi, "Calcul approché de la série d'Iwasawa pour les corps quadratiques (p = 3),"

Number Theory, 1981-82 and 1982-83, Exp. No. 4, 68 pp., Publ. Math. Fac. Sei. Besançon, Univ.Franche-Comté, Besançon, 1983.

7. H. W. Leopoldt, "Eine Verallgemeinerung der Bernoullischen Zahlen," Abh. Math. Sem. Univ.Hamburg, v. 22, 1958, pp. 131-140.

8. T. Metsänkylä, "Iwasawa invariants and Kummer congruences," /. Number Theory, v. 10, 1978,pp. 510-522.

9. I. Sh. SlavutskiI, "Local properties of Bernoulli numbers and a generalization of the Kummer-Vandiver theorem," Izv. Vyssh. Uchebn. Zaved. Mat., No. 3 (118), 1972, pp. 61-69. (Russian)

10. S. S. Wagstaff, Jr., "The irregular primes to 125000," Math. Comp., v. 32,1978, pp. 583-591.11. L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, Berlin and New York, 1982.


Documents

a-= E K · 2018. 11. 16. · where \pk = \pu~k and Bk(^k) stands for the Arth generalized Bernoulli number attached to the character \pk. Denote by Q^x) the extension of Qp generated