CLASS NUMBERS of ABELIAN NUMBER FIELDS Jacques Martinet Has Sketched the Development of the Main Ideas in the Theory of Cyclotom

CLASS NUMBERS OF ABELIAN NUMBER FIELDS

FRANZ LEMMERMEYER

Jacques Martinet has sketched the development of the main ideas in the theory of cyclotomic fields from the publication of Hasse’s monograph up to the heydays of Iwasawa theory (for a polished proof of the Main Conjecture see e.g. the beautiful book [10]). The notorious conjectures (Vandiver, Greenberg) are still open, and progress has shifted somewhat to the nonabelian side. This note is devoted to the algorithmic part of the theory of cyclotomic fields. As Hasse explained in his preface, he was not content with the state of the art in the theory of cyclotomic fields: Hilbert had erected a magnificient building of algebraic number theory, which had been crowned by Takagi’s class field theory. On the other hand, the computiation of the main invariants of number fields, such as the unit group and the class number, was not feasible except for fields of small degree and very small discriminant. In Hasse’s opinion it was unsatisfactory to have such a great tool as the analytic class number formula while having to admit that it was pretty useless for actually computing class numbers. Hasse’s goal was to investigate the class number formula from an arithmetic point of view; even showing that the number h− provided by the class number formula is an integer is quite nontrivial. He then showed that the class number formula may be used for computing the relative class numbers of all cyclotomic fields of conductor ≤ 100. Hasse’s monograph served as a blueprint for the corresponding book on abelian extensions of complex quadratic number fields by Curt Meyer [46]. The whole situation in this case is much more complicated, and Meyer’s book is a lot harder to read than Hasse’s. What is missing are the many tables that Hasse provided in his book, and in particular the beautiful diagrams of the subfields involved.

Class Number Formulas. The statement that the analytic class number formula for quadratic number fields was first proved by Dirichlet must be taken with a grain of salt, since Dirichlet worked with binary quadratic forms. This fact makes trans-√ lating his result on the class number formula for Dirichlet number fields Q(i, m ) (or, as he would have said, for binary quadratic forms with complex coefficients and determinant m) a highly nontrivial task, since his class groups of forms correspond to certain ring class groups in the modern sense. The generalizations of his class number formula to more general biquadratic number fields provided by Eisenstein [15], Bachmann [5] and Amberg [1] are so difficult to translate into the language of number fields that it is hard to say whether their results are correct or not (Hasse alludes to this state of affairs in his Introduction). Hilbert [27] and Herglotz [26] proved these results in the language of number fields, and the most profound study of the class number formula in multiquadratic number fields is the little known thesis by Värmon[69], which contains most of the results that were later rediscovered independently by other authors. 1 2 FRANZ LEMMERMEYER

Computation of Class Numbers. Already Gauß had computed extended tables of class numbers of binary quadratic forms, but he used his theory of composition, a method that is superior to the class number formula for computations by hand. Even after Dirichlet’s proof of the class number formula in the quadratic case, the theory of quadratic forms remained the favorite computational tool. Similar techniques were not available when Kummer started investigating the class groups of cyclotomic fields. He proved that the class number h of the field + − + K = Q(ζp) of p-th roots of unity admits a factorization hp = hp hp , where hp + −1 is the class number of the maximal real subfield K = Q(ζp + ζp ), and where − − hp is an integer that can be computed explicitly. The relative class number hp = h(K)/h(K+) is numerically accessible, the plus class number is essentially the index of the group of cyclotomic units inside the full unit group. Using the techniques provided by Kummer, C.G.Reuschle [57] (1812–1875), a teacher at the Gymnasium in Stuttgart, computed generators of the principal ideals of small norms in cyclotomic number fields Q(ζm) for all prime values m < 100 as well as for several composite values of m ≤ 120. Reuschle’s correspondence with Kummer was published by Folkerts and Neumann [17]. After the publication of Hasse’s book, Schrutka von Rechtenstamm [64] published extensive tables of relative class numbers of cyclotomic fields in 1964. D. H. Lehmer [37] computed many minus class numbers of cyclotomic fields; his results were extended considerably in [18]. Yoshino and Hirabayashi [72] expanded Hasse’s tables including the diagrams for the subfields (see the appendix of the present translation). + The first calculations of hm in some nontrivial cases were done by van der Linden + [40], who used Odlyzko’s bounds for computing hp for all primes p ≤ 163 (in some cases he had to assume GRH). Great advances were made by R. Schoof [62] (see also [33] and [22]), who was able to determine factors of class numbers of real cyclotomic fields, which very likely coincide with the actual class numbers. StéphaneLouboutin has written a wealth of articles on the computation of relative class numbers of CM-fields using analytic means, and has used this techniques for classifying abelian (and non-abelian) CM-fields with small class numbers; see e.g. [41]. Class numbers of cyclotomic fields showed up in investigations of Catalan’s equation xm − yn = 1. It is not difficult to reduce the statement that this equation does not have any nontrivial solutions to the case xp − yq = 1, where p and q are distinct odd prime numbers. It can be shown (see [7, 47]) that if this equation has a − − nontrivial solution, then p | hq and q | hp . The conjecture that the only nontrivial solution of the Catalan equation in natural numbers is 32 − 23 = 1 was obtained by P. Mihailescu using Stickelberger’s theorem (see [6, 63] for expositions of the proof). For an overview on the determination of class numbers using the p-adic class number formula see the recent thesis of Zhang [73].

Parity of Class Numbers. Hasse proved in Theorem 45 that the class number of − − p−1 Q(ζn) is odd if and only if hn is odd. The conjecture that hp is odd if p and q = 2 are both prime emerged in the work of Davis [13] and Estes [14]. This conjecture was proved if 2 is a primitive root modulo q by Estes; see Stevenhagen [66] for a CLASS NUMBERS OF ABELIAN NUMBER FIELDS 3 generalization and other useful references. Gras [21] proved an important duality result between groups of totally positive and primary cyclotomic units; a diﬀerent proof based on the Main Conjecture was given recently by Ichimura [30]. Yoshino [71] proved that the class number of Q(ζn) is even if n 6≡ 2 mod 4 is divisible by 4 distinct prime factors. For the state of the art concerning the parity of plus class numbers, and the determination of the 2-class group of real abelian ﬁelds, see the recent thesis [70] by Verhoek.

Structure of Class Groups. Already Kummer [34] investigated the structure of some minus class groups of cyclotomic fields by studying the action of the Galois group on class groups; in this way he was able to show that the minus class group of Q(ζ29), which has order 8, is elementary abelian. Subsequently, Kummer’s methods were refined by Tateyama [67], Horie & Ogura [29], and many other authors. + − Kummer’s result that p | hp implies that p | hp was proved algebraically by − Hecke [24], who proved that the p-rank e of the minus part of Clp(Q(ζp)) and the + − + corresponding rank e of the plus part satisfy e ≥ e . Generalizing results√ by Arnold√ Scholz [60], who compared the 3-ranks of the ideal class groups of Q( m ) and Q( −3m ), Leopoldt [39] obtained strong bounds between individual pieces of the plus and the minus class groups of cyclotomic fields.

Class Number 1 Problems. Kummer conjectured in 1851 that the minus class − number hp of Q(ζp), where p is prime, is asymptotically equal to

p+3 p 4 G(p) = p−3 p−1 . 2 2 π 2 Ankeny and Chowla could prove that h− log p = o(log p), G(p) − which implies that hp = 1 for only finitely many primes numbers p. Granville [20] showed that Kummer’s conjectures is not compatible with conjectures in analytic number theory that are believed to be true. Masley [42] (see also [43]) determined all cyclotomic fields Q(ζm) with class number 1 and found that all of them satisfy m ≤ 84. A useful result in this connection was proved by Horie [28]: if K ⊆ L are complex abelian number fields, − − then hK | 4hL ; this was generalized to arbitrary CM-fields by Okazaki [54]. In this book, Hasse discusses Weber’s result that the class numbers of the fields n+2 Ln of 2 -th roots of unity are all odd in § 34. Harvey Cohn [12] pointed out that the maximal real subfields Kn of Ln seem to have class number 1. This is easy √ p √ to prove for K1 = Q( 2 ) and K2 = Q( 2 + 2 ), and it follows from Reuschle’s tables that h(K3) = 1. Cohn proved that the fields Kn either have class number 1 or ≥ 257. He adds the remark that “We still have obtained no evidence to doubt” that the class numbers of the fields Kn are all trivial. Fukuda and Komatsu [19] improved Cohn’s result and showed that the class 9 numbers h(Kn) are not divisible by any prime number < 10 . Bauer and van der Linden showed that h(K4) = h(K5) = 1, Miller [48, 49] proved that h(K6) = 1 and, assuming GRH, that h(K7) = 1. In addition, he conjectures that all subfields Kp,n of the cyclotomic p-extensions for any prime p and any n have class number 1. 4 FRANZ LEMMERMEYER

Buhler, Pomerance and Robertson [8] have shown that the Cohen-Lenstra heuristics predict that h+(pn) = h+(p) for almost all primes p and all integers n, where h+(pn) is the class number of the maximal real subﬁeld of the ﬁeld of pn-th roots of unity, and they remark that “it is possible that there are no exceptional primes” p at all.

Hilbert Class Fields. The fields Q(ζp) with p ≤ 19 have class number√ 1, and L = Q(ζ23) has class number 3 coming from the quadratic subfield K = Q( −23 ). In particular, we get the Hilbert class fields of L and K by adjoining a root of the polynomial f(x) = x3 − x + 1 with discriminant −23. In a similar way we can construct unramified cubic extensions for many other cyclotomic fields. The smallest example of a quadratic number field with class number 5 was treated by Hasse [23]. Nowadays such calculations can be performed routinely using the methods described in Henri Cohen’s book [11]. A few other known examples of unramified abelian extensions of cyclotomic fields are given by the following table.

∆ h f | disc F | 47 5 x5 − 2x4 + 2x3 − x2 + 1 472 79 5 x5 − 2x4 + 3x2 − 2x + 1 792 71 7 x7 − 2x6 + 2x5 + x3 − 3x2 + x − 1 713 29 8 x8 − 4x7 + 8x6 − 6x5 + 2x4 + 6x3 − 3x2 + x + 3 296 31 9 x9 − x7 − 2x6 + 3x5 + x4 + 2x3 − x2 + x − 3 316

The field F of degree 17 whose compositum with K = Q(ζ64) is the Hilbert class field of K was computed by Noam Elkies [16]: F is generated by a root of the polynomial f(x) = x17 − 2x16 + 8x13 + 16x12 − 16x11 + 64x9 − 32x8 − 80x7 + 32x6 + 40x5 + 80x4 + 16x3 − 128x2 − 2x + 68 and has discriminant | disc F | = 279. Families of unramified extensions of cyclotomic number fields were constructed by Arnold Scholz ([61]; see also [38]). An investigation of Metsänkylä’sresults [45] on prime factors of the minus class number of cyclotomic fields from a class field theoretical point of view might be a rewarding project. The main investigation of unramified abelian extensions of cyclotomic fields was done in connection with p-class groups of Q(ζp). Kummer had already shown that these p-class groups are governed by the divisibility of Bernoulli numbers by p, and some of his results may be interpreted as an explicit construction of p-class fields of Q(ζp). The precise connection between the divisibility of Bernoulli numbers and certain pieces of the p-class group of Q(ζp) was investigated by Pollaczek [55], Morishima [51, 52] and Vandiver [68], but the clearest exposition and the most complete results concerning this correspondence were given by Herbrand [25]. For K = Q(ζp), set p a G = Gal(K/Q), A = Cl(K)/ Cl(K) and σa(ζ) = ζ . Let

ai Ai = {c ∈ A : σa(c) = c for all σa ∈ G.}. CLASS NUMBERS OF ABELIAN NUMBER FIELDS 5

Then A0 is the part of A ﬁxed by the Galois group, and we have

A = A0 ⊕ A1 ⊕ · · · ⊕ Ap−2.

It is easy to see that A0 = A1 = 0. Herbrand proved that if Ai 6= 0 for some − odd index i, then p | Bp−i, which refines Kummer’s result that if p | hp , then p divides some Bernoulli number. Since the plus part of the class group is the sum of the Ak with even index, Vandiver’s conjecture states that Ak = 0 for all even 0 ≤ k ≤ p − 3. Herbrand proved that if p | Bp−i, then Ai 6= 0 if Vandiver’s conjecture holds, and Ribet [58] (see also Mazur’s beautiful survey [44]) succeeded in eliminating Vandiver’s conjecture using modular forms. In recent years it was discovered how to use algebraic K-theory for proving results about the Ai. Kurihara [35] was able to show Ap−3 = 0, and Soulé[65] that 224n4 Ap−n = 0 for odd values of n satisfying log p > n . The triviality of certain pieces of A is also related to a conjecture of Ankeny, √ Artin and Chowla [2, 3, 4], according to which the fundamental unit ε = t + u p for primes p ≡ 1 mod 4 satisfies p - u. Kiselev [31], Carlitz [9] and Mordell [50] proved independently that this is equivalent to p - B(p−1)/2. This conjecture was verified for all primes p < 2 · 1011 in [56]. For a survey on the fascinating connections between values of zeta functions and algebraic K-theory see Kolster [32].

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