Cyclotomic Invariants and ^-Irregular

Transcription

1 MATHEMATICS OF COMPUTATION, VOLUME 32, NUMBER 142 APRIL 1978, PAGES Cyclotomic Invariants and ^-Irregular Primes By R. Ernvall and T. Metsänkylä Abstract. We prove some general results about the Iwasawa invariants \ and u of the 4pth cyclotomic field (p an odd prime), and determine the values of these 4 invariants for p < 10. The properties of A. and u are closely connected with the i-irregularity (i.e. the irregularity with respect to the Euler numbers) of p. 4 A list of all i-irregular primes less than 10, computed by the first author, is included and analyzed. 1. Introduction. Let p be an odd prime. For a natural number m prime to p, consider the p-class groups of the cyclotomic fields Kn of mpn+lth roots of unity (n 0, 1,... ). For all sufficiently large n, the orders of these groups equal pe^ with e(n) = Xrc + pp" + v, where X = X and p = pm are nonnegative integers that (as well as v) do not depend on n. The same holds true when Kn is replaced by its maximal real subfield; let us denote then the corresponding invariants by X+ and p+. Put À = X- + X+, p = pt + p+. Then the invariants X- = X~ and p~~ = /u~ are related to the exact power of p dividing the first factor h~ of the class number of*. Iwasawa [10] has conjectured that p = 0 for every choice of m. This has been proved only for p = 3 [5]. Note that X+ < X-, p+ < p~ (see e.g. [5, p. 63]) so that the results X = 0 and p = 0 are implied by X- = 0 and p~ = 0, respectively. We also know that X- = p~~ = 0 if and only if p does not divide h~\ /h^ (see [9, p. 95], where X- and p~~ axe denoted by X and u). Suppose that m = 1. Then the condition X~ = p~ = 0 is also equivalent to the fact that p is a regular prime [9, p. 96], i.e. p^h~ü or, equivalently, p does not divide the numerator of any of the Bernoulli numbers B2, B4,..., B 3. For irregular primes p, the invariants X~ and p~ (and X, ju ) have been determined with the help of computers up top < 125,000 [11], [14], [23]. It has turned out that pp = 0 for all these p. In this paper we shall be concerned with the case m = 4. Although this case is rather similar to the case m = 1, some new features appear. We shall prove that p4 = pz if p is E-regular, i.e. p does not divide any of the Euler numbers E2, E4,, Ep_3- Furthermore, using results obtained by computer, we shall show that p4 = 0 for every prime p < 104, and determine the value of X^ for these p. Received February 8, 1977; revised August 15, AMS (MOS) subject classifications (1970). Primary 10A40, 12A35, 12A50; Secondary 10B15. Key words and phrases. Class numbers, cyclotomic fields, Z -extensions, E-irregular primes, irregular primes, Euler numbers, Fermat's Last Theorem, Fermât quotients. 617 Copyright 1978, American Mathematical Society

2 618 R. ERNVALL AND T. METSANKYLA We note that the connection between the "-regularity of p and the divisibility by p of the class number of the 4pth cyclotomic field was discovered by Gut [7] (see also [21]). Gut [6] has also found that the -regularity of p is connected with the solvability of the diophantine equation x2p + y2p = z2p. The list of -irregular primes produced by our computation procedure is also interesting in its own right. This list is included at the end of this paper and analyzed in Section 2. Among other things, it may be compared with the corresponding list of ordinary irregular primes (called B-irregular below). Section 2, together with Section 7 containing a report of the computations, is due to the first author, who also prepared the computer programs. Sections 3-6 concerning X4 and p~^ are work of the second author. 2. Irregular Primes. Euler numbers En (n = 0, 1,... ) can be defined by the symbolic equations (E+ l)n +(E-l)n = 2 foxn = 0, = 0 for n > 1 (see, e.g. [19, p. 25]). It follows that all the En are integers and those with an odd index equal zero. Moreover, (1) (m +E+ l)2n +(m + E- l)2n = 2m2" (n > 1), where m is an arbitrary integer. Take an odd k. Letting m run through odd integers from 1 to 2k - 1, we get from (1) 2k (2) E2n = ^d(a)a2" (mod*2), a= I where 0 is the unique Dirichlet character with conductor 4. Almost all the properties of Euler numbers needed in the sequel are based on this congruence. We say that a prime p > 5 is E-irregular if there exists an even integer 2n such that 2 < 2m < p - 3 and p divides E2n. We then say that (p, 2«) is an E-irregular pair. For each p, we call the number of such pairs the index of E-irregularity of p and denote it by ie. Carlitz [3] has proved that there are infinitely many "-irregular primes. The first author [4] has shown that the number of the -irregular primes P ±1 (mod 8) is infinite. primes. It is not known whether there are infinitely many E-regular We used a computer to find all -irregular pairs (p, 2n) with p < 104. The table at the end of the paper lists all these pairs. There are 495 -irregular primes in all. It should be noted that, as was to be expected, they are quite evenly distributed mod 8. Furthermore, ie = 2 for 86 primes and ie = 3 for 15 primes. The case ie = 4 occurs for the primes 3673 and 8681 and the case ie = 5 for No prime with ie > 6 was found. (For these and the following results, compare the corresponding results concerning 5-irregular primes [12], [14], [23].) Gut [6] has proved that the condition 3 = E 5 s = = E x, = 0 (mod p) is necessary for the equation x2p + y2p = z2p (p\xyz) to be solvable. Vandiver [22] has given a proof of the fact that if xp + yp = zp (p-[xyz) is satisfied,

3 CYCLOTOMIC INVARIANTS AND ß-IRREGULAR PRIMES 619 then (p, p - 3) is an -irregular pair. In our range we found that (p, p - 3) is an -irregular pair for p = 149 and 241, (p, p - 9) is such a pair for p = 19, 31, and 3701, and (p, p - 11) for p = 139 only, while there is no example of an -irregular pair of the form (p, p - 5) or (p, p - 7). No consecutive -irregular pairs (of the form (p, 2n) and (p, 2n + 2)) were found. For each -irregular pair (p, 2n) we also computed 2n mod p2. It appeared that E2n is never divisible by p2 for p < 104 (cf. [13], [14], [23]). Denote by irb(x), tie(x), and irbe(x) the number of those primes, not exceeding x, which are 5-irregular, -irregular, and both B- and -irregular, respectively. Siegel [20] predicted that the ratio tt b(x)/tx(x) approaches the limit 1 - e'1^2 = as x -* «x». This result can be obtained by assuming that the numerators of the Bernoulli numbers B2, B4,..., Bp_3 axe randomly distributed mod p. The same hypothesis on the Euler numbers 2, E4,..., p_3 leads to the conjecture that tte(x)/tt(x) -> 1 - e-x>2 and n BE(x)lit(x) -* 1-2e"1/2 + e_1 = as x -> oo. The information obtained from our computations seems to support these hypotheses, as is seen from the following table. (The values of ttb and nb/it axe appended in this table for the sake of comparison. In calculating nb and nbe we used the table computed by Johnson [14].) 'B '"be ïïb/1î Te/7* *be^ As in the case of 5-irregular primes, one is also led to the conjecture that the -irregular primes with index k satisfy the Poisson distribution rke-i/fc! with t = la. The table below compares the actual number of primes of each index within our range with these predictions. Index Observed Expected 0 1 > Total Preliminaries About the Iwasawa Invariants. We shall treat the invariants X4 and U4 on the basis of the theory of p-adic -functions, due to Iwasawa [9, Section 6]. Denote by Z the ring of p-adic integers. For a rational integer a prime to p, let io(a) E Z be the p-adic limit of the sequence {ap }. Then (3) u(a) = ap" (mod p" + 1Zp) for all n > 0, and co can be viewed, in a natural way, as a Dirichlet character that generates the character group mod p.

4 620 R. ERNVALL AND T. METSANKYLA For each n > 0, let on(a) denote the residue class mod 4p"+1 determined by the integer a, and put r = {0»lfl = l (mod4p)}, A = {on(a) I a odd and a""1 = 1 (mod p" + 1)}. It is easy to verify that the multiplicative residue class group mod 4p"+1 is the direct product of its subgroups Fn and A. Denote by An the set of integers a with 1 < a < 4p"+ ' and (a, 4p) = 1. Fix p" (= the order of Tn) integers cn so that 1 < cn < 4pn+l and, for each a EAn, on(a) = on(cn)on(dn), on(cn) E T, on(dn) E A. In the following x will denote an even character whose conductor f equals p or 4p. Let R be the inverse limit of the group algebras Z [Tn] with respect to the natural homomorphisms, induced by om(a) h» on(a) (m>n). (4) ï = Ç (x) = - (8p" + 1r' Z ax(a)orl(a)on(cnrl. a<ean We know that %n E Zp[r ] For n > 0, write and that = lim %n is a well-defined element of R (see [9, pp ], where on(cn) and on(dn) are denoted by y (a) and 5n(a), respectively). Moreover, there exists an isomorphism r from R onto the formal power series algebra Zp[[x]] such that the image of % under t, say /0;x)= f akxk ezp\\x]], k=0 has the following connection with the p-adic -function L (s; x): Lp(s: X) = 2/(0 +4p)*-l;X) for all sez (see [9, pp. 69, 77] ). This implies, among other things, that f(x; x) does not vanish identically. X(x) and p(x) such that Consequently, there are unique nonnegative integers fix; X) = PM(X) t bkxk (bk E zp) fc=0 with bk = 0 (mod pzp) for 0 < k < X(\) and bhx) 0 (mod pzp). Then (5) X4~p=X>(x), M4P=Za'(x), X X where x ranges over all even characters with / = p or 4p [17, p. 65]. Let X stand for the set of all even characters with conductor 4p, that is, We rewrite the equations (5) as X = {6u>m + 1 \m even and 0 < m < p - 3 }.

5 CYCLOTOMIC INVARIANTS AND E-IRREGULAR PRIMES 621 (6) \p = \ + Z *(X), P4p = Hp + Z M(X)- To obtain information about X(x) and p(x) we have to investigate the divisibility by p of the coefficients a0, ax,... of f(x; x). For this purpose we need a relationship between f(x; x) and the generalized Bernoulli numbers (for the definition of these, see,e.g. [9, p. 9]). Indeed, for n > 1 and x^iwe have (7) 2/((l + 4p)1-" - 1 ; X) = - (1 - (x^n)ip)pn'l)bn(xorn)/n, where Bn(\jj) denotes the nth generalized Bernoulli number belonging to the character \p [9, p. 78]. Below we shall employ this formula for n = 1 and n = 2 only; then it will be useful to know that (8) BXW) =/-' Z *(«>» 0=1 (9) B2w)=rx z ^y a=l (* odd), (\p even), where / = /)// > 1 ([9, p. 14] and [17, p. 67]). In studying X(x) and p(x) for x = 0a>m + ' E X we have to distinguish between the cases m = 0 and m The Zero Case. Theorem 1. p(qos) = 0. Proof. Write the formula (4), for x = öco, in the form K=HSn(cn)on(cnT\ V^) = -(8p"+1r1 Z ab(a), c» a^an(cn) where cn ranges over all its p" values and An(cn)={aEAn \on(a)eo (cn)an}. Assume that p(9cj) > 0. From a result proved in [17, p. 69], we then infer that Sn(cn) = 0 (mod PZp) for all n > 0 and all cn. Let a E An with 1 < a < 2p"+1. Then it is seen that a E An(cn) if and only if a + 2p" + 1 EAn(cn). Indeed, suppose that a EAn(cn); there is an integer dn such that a = c dn (mod4p"+1), on(dn) E A, and then a + 2p"+1 =-c (J + 2p"+1) (mod 4p"+1), where, furthermore, on(dn + 2p"+1) G A. The converse is verified by a similar argument. Observing that B(a + 2p"+1) = - 6(a) we thus obtain

6 622 R. ERNVALL AND T. METSANKYLA S (cn) = - (8p"4"1)-1 5>0(«) + («7 + 2p" + 1)0(a + 2p"+1)] = 4~l^e{a\ a " where the sums are extended over those numbers a EAn(cn) for which 1 < a < 2p"+1. The last sum consists of p - 1 terms 8(a) = ±1. Being divisible by p, it must therefore vanish. Consequently, = 0 for all n > 0. This in turn implies that? = 0, and so /(x; 0co) = 0, which is a contradiction. Theorem 2. Ifp = 3 (mod 4), then X(0co) = 0. If p = 1 (mod 4), //Acti X(0gj) > 0. Proof. By setting «= 1 in (7) we get a0=f(0;9co) = -(\-9(p))Bx(9)/2. It follows from (8) that Bx(0) = - la. Hence a0 = la if p = 3 (mod 4), and a0 = 0 if p = 1 (mod 4). In view of Theorem 1 this proves our assertion. The proof of Theorem 2 also gives an easier proof of Theorem 1 in the case p = 3 (mod 4). As a consequence of Theorem 2, one finds that X4 > 0 if p = 1 (mod 4). We remark that the weaker result X4 + p~^ > 0, for p = 1 (mod 4), follows also directly from [16, Satz 10] which concerns the divisibility by p of the first factor of the class number of the 3p"+1th and 4p" + 1th cyclotomic fields. Theorem 3. Let p = 1 (mod 4). Then X(0cj) > 1 if and only if the Euler number E x is divisible by p2. Proof. Since in this case /(x; 0co) = axx + a2x and p(0co) = 0, the condition X(0co) > 1 is equivalent to p I ax. Put a = (1 + 4p)_1-1 = - 4p(l + 4p)_1. Equation (7) gives, for n = 2, the relation 4/(a;0co) = -B2(6oT1). Accordingly, p ax if and only if B2(BoTl) = 0 (mod p2zp). We shall show that (10) A2(0co-1)= p_1 (mod p2zp); by the above this proves the theorem. Using (9), we obtain B2(8oTl) = (4p)-1 Z e(a)gt1(a)a2 a=l = - Z d(a)oj~1(a)a-p 9(a)oT\a). 0=1 0=1 The last sum here vanishes, as 0(2p - a) 9(a) and co(2p - a) = - co(a). By (2) and (3) we, therefore, see that (10) is equivalent to 2p Z 9(a)(cJ-l(a)a + G)(a)a~l) s 0 (mod p2zp). 0=1 The validity of this congruence follows from the identity

7 CYCLOTOMIC INVARIANTS AND E-IRREGULAR PRIMES 623 2p Z 6(a)oJ-l(a)a(l~co(a)a-1)2=0 (mod p2z ) 0=1 p on noting that p = 1 (mod 4) implies Z2fLx9(a) = 0. Hence, our theorem is proved. Note that for p = 1 (mod 4), E x is always divisible by p. This can be seen either from the preceding proof or, of course, directly from (2). - On checking by computer all primes p less than 104 and congruent to 1 (mod 4), we found that j was never divisible by p2. Hence, we have the result: if p = 1 (mod 4), then X(0w) = 1 whenever p < The Remaining Cases. In the following the statement m#0 will mean that m is even and 2 < m < p - 3, so that the character x = 0com + 1 belongs to X and is different from 0co. It should be noted that the considerations in this section (as well as above in the proof of Theorem 3) are partly similar to those presented in [17, Section 6], where the case of the pth cyclotomic field was discussed. Theorem 4. Ifx = 9com + l, m ^ 0, then X(x) = p(x) = 0 if and only if the pair (p, m) is E-regular. Proof. The same arguments as before give now 4a0 = - 2BX(9<J") = - (2p)-1 0=1 6(a)um(a)a = Z 0(a)œm(a) = Em (mod pza 0=1 On the other hand, p^a0 if and only if X(x) = p(x) = 0. congruence Theorem 5. Let x 0com + 1, m^o. If the pair (p, m) is E-irregular and the (11) Em=Em+p_x (modp2) does not hold, then X(x) = 1 and p(x) = 0. Proof. By Theorem 4, it suffices to show that p \ax implies (11). Let p\ax and choose a as in the proof of Theorem 3. Then Since /(O;0«m + 1)=/(a;0wm+1) (mod p2zp). 4f(a; 0com + 1) = - B2(e^m~l) = ^ 9(a)um-l(a)a, 0=1 the above congruence can be written in the form This yields 9(a)œm-1 (a) (( (a) - a) = 0 (mod p2zp). 0=1 2r> Z 9(a)am~l(ap -a) = 0 0= 1 (mod p2zp),

8 624 R. ERNVALL AND T. METSANKYLA and so the assertion (11) follows by virtue of (2). Our computer search indicates that (11) does not hold for any -irregular pair (p, m) with p < 104. Consequently, X(0com + 1) = 1 and p(9com + 1) = 0 whenever m = 0 and the pair (p, m) is E-irregular with p < 104. An inspection of the preceding proof shows that one can prove even more, namely that p ax is equivalent to (11). Put Kr(2n)=t(-iy(r)E2n + i(p:x) (, = 0,1,...). f=o Vi/ Then (11) can be written as Kx(m) = 0 (mod p2). By Kummer's congruences [18, Chapter XIV], Kr(2n) = 0 (mod pr) for 2«> r, so that the preceding congruence is always true mod p. If (11) did hold for some -irregular pair (p, m), it would be rather easy to check whether p \a2 or not Indeed, the second author has proved that generally a0 = ax =... = ak = 0 (mod pz ) if and only if Kr(m) = 0 (mod pr+1) for r = 0,..., k, provided that k <m. The proof will appear elsewhere. 6. Summary of Results About the Iwasawa Invariants. Summarizing the results from Theorems 1-5 and from our computer search we may state, by (6), that (i) P4 = p~ if either p is -regular or p is -irregular and less than 104 ; 00 ^4P = Xp if p = 3 (mod 4) and p is -regular; (iii) X-p = Xp + ie if p = 3 (mod 4) and p < 104 ; (iv) \4 = Xp + 1 if p = 1 (mod 4) and p is -regular; (v) X~p = Xp + ie + 1 if p = 1 (mod 4) and p < 104. Note in this connection that, by known results, pz = XI = 0 if and only if p is irregular, and up = 0 and Xp equals the index of 5-irregularity of p for all 5-irregular primes less than 125,000 (see [14], [23]). 7. The Computations. All the computations were performed on the UNIVAC 1108 computer at the University of Turku, and they took about 26 hours. Only integers and vectors consisting of integer components were used, and so the possibility of round-off errors was avoided. We used the following criterion in order to find out the -irregular pairs (p, 2n). The powers of integers needed here were calculated by the aid of a primitive root g, which was computed first (the values of g were checked from a table). Theorem 6. A necessary and sufficient condition for (p, 2ri) to be an E- irregular pair is that l2" + 22" [p/4]2" s 0 (mod p). Proof. Put s = (p- l)/2. By (2), 2 =-202"-32" (-iy-1(p-2)2"} = (-\y-l2{22" -42n (-l)s-1(p- l)2"} = (_1y-i22«+i{12«_22«+_ +(_,y-is2n} (modp)_

9 CYCLOTOMIC INVARIANTS AND E-IRREGULAR PRIMES 625 Table E-irregular pairs (p, 2«) to p < 104 2n 2n 2n 2n

10 626 R. ERNVALL AND T. METSANKYLA Table (continued) 2n 2n 2n 2n

11 CYCLOTOMIC INVARIANTS AND E-IRREGULAR PRIMES Table (continued) 627 2n 2n 2n 2n

12 628 R. ERNVALL AND T. METSANKYLA On the other hand, Combining these congruences, we see that 2 Z k2n~ Z^2"=0 (modp). fc=i fe=i E2n =(- I)i24" + 2{12" + 22" [p/4]2"} (modp). This proves the theorem. For each -irregular pair (p, 2m) we computed 2 /p modp and ( 2 +p_i ~E2n)/p mod p on the basis of the congruence (2). to compute E Similarly, this congruence was employed x/p mod p for each p = 1 (mod 4). To write (2) in a more suitable form, observe first that the Fermât quotient of an integer u prime to p is defined as the least nonnegative integer qu satisfying the congruence up~l = 1 + qup (mod p2). It is easy to verify that q2p_u Qu + 2wp_2 (mod p). Hence (2) yields the following congruences which were actually used in the computations: E2n/p = 2p-1 (- l)*+»(2* - l)2" -4n Z (- 1)*+1(2* - l)2""1 (mod p), k=l k=l (E2n+P-i - E2n)/P =- 2 Z (- 0*+1 i(2fc - l)2"fl2fc-i + (2* - 1)2""1 } (mod P). fc=l p-i/p = 2 Z (- 0fc+ ' {i2fc_! + (2* - If"2 } (mod p) (for p = 1 (mod 4)). fc=i As a check, we computed the value mod p of the expression 5 = -6 Z (2k-\fq2k_l. k=l Indeed, it follows from the known congruence p-i B2 + 2 Z fc=i fc2<7k = 0 (mod p) (see, e.g. [15, p. 255] ) that S = 1 (mod p). A further check was supplied by the value of q2, which was also printed and then compared with Haussner [8]. Our value of q2 was different from that of [8] for eleven primes, namely 2437, 4049, 4733, 4969, 5689, 6113, 6997, 7121, 7321, 8089, and A comparison with Beeger's tables [1], [2] showed that in these cases q2 is incorrectly given in [8]. We note that there can be more errors in [8] (extended up to the prime ), for we checked only the primes (< 104) which are either congruent to 1 (mod 4) or -irregular. A table including all the results of our computations has been deposited in the UMT file. Department of Mathematics University of Turku SF Turku SO, Finland

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