Heuristics on class groups

Heuristics on Class Groupa by H. Cohen and H.W. Lenstra, Jr. 81 Experlmental facts. This work was motivated by the desire to understand eorae experi- mental observations about class groups of quadratic fields; these observations were made long ago, and confirmed by the raoat extensive tablee known to us, I.e. Buell [1] for iraaginary quadratic fields, and Shanks and Williams [5] for real guadcatic fields. They are äs followsi (A) If p is a small odd prime, the proportion of imaginary quadratic fields whose class number h(o) is divisible by significantly greater than l/p. p is For example for p = 3 it is around 43% instead of the expected 33.34,, for p = 5 around 23. 5< instead of 20<< etc. (B) If one looks only at the odd part of the class group, cyclic groups eeem to form the overwhelming majority. In fact, it is quite difficult to find class groups with, say 3-rank greater than or equal to 3, and no examples are known o£ 3-rank greater than 5. (C) For real quadratic fields <ü(Jp) with p a l mod 4 prime, it is conjectured than an infinite number of them have class number l, and in fact experimental evidence seems to ahow that there is a definite nonzero proportion p of fields (D(,/p) of class number 1. A rough extrapolation of known data seems to give p *, 764. The work we are about to describe gives quantitative heuristic explana- 27 tions of all these obaervations and of many more, Including for higher degree fleide. Por example, for Imaginary quadratic fielda we predict that p|h(D) with approximate probabllity 43.987< for p - 3, 23.9674, for p = 5. For real quadratio fields the Proportion p of class nuraber l ahould be approximately 75.4464. A more detailed Version of this paper including complete proofa and extra material will be published elsewhere. 82 Heuristic aasumptiona The baaic clue comea from experimental fact (B). Let us consider a specific examplet assume that the 3-part of the claas group of an imaginary quadratic field has cardinality 9. only the two groupe Z/9Z and (Z/31) that (Z/3Z) Then up to iaomorphism, can occur. occurs much more rarely. However, tables ahow Why? One anawer is that the automorphism group of a cyclic group ia amaller than the automorphism group of any other abelian group of the same cardinality. *Aut(Z/3Z) = 48. In our example, *Aut z/9Z « 6 while The basic heuristic aasumption is thus that isomor- phism classea of abelian groups l/jfAut G. G have a "weight" proportional to This ia similar to many "masa formulaa" in other parts of mathematicB where the proper v.'eight is indeed the inverae of the number of automorphisms. For our example thie agrees quite well with the tables aince the ratio of occurrence of Z/9Z versus (Z^Z) ia close to 8 to 1. iVnother way of statlng our asaumption is äs followa: an abstract set with n elements. let E be It can easily be shown that the number of abelian group structurea on E which are isomorphic to G 28 ia equal to n.'/Mut G. Hence for a given Order n, weighting isomorphiam classes of abelian groups with veight proportional to 1/^Aut G is äquivalent to giving equal weight to each abelian group structure. We are thua led to the following assumption. Let f be a function defined on the isomorphiam classes of finite abelian groups of odd order. We define the average of M(f) - lim uhere D l0!^ f by '°ää- if the lirait existe, goes through the sequence of negative fundamental discrimi- nants,zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ίί(0)zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ia the class group of BKv^D), and for every abelian group G, G o d d denotea the odd part of tion of a property G. If f is the characteristic func- 9, we call M(f) the probability that 9 holds. The assumption ia then äs f öl Iowa j Heuristic aasumption 1t M(f) Χ·*·» Σ4 where the sums are to be taken over isomorphiam classes of abelian groups G of cardinality lese than or equal to x, Remarka 1) We reatrict to the odd part of the class groups aince the even part is certainly not random because of genus theory and the theory of atnbiguous classes. 2) It could very well be argued that one could replace the weighting factor l/#Aut G by a factor of the form |(#G)/*Aut G, where t is a smooth function. However it can be shown that for a very wide 29 claas of functions i including for instance the nonzero polynoraials, the limit of is independent of t, so therezyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM ία not much loss in generality in assuming ψ = 1. 83 Some algebraic and analytic reaults To be able to use our heuristic asaumption l above, we need to have a number of algebraic and analytic resulta, The key algebraic result which we need is the followingt Theorem 3. l ι Let K and C be finite bbelian groups. Then ΐ tiG·. subgroup of Gi G L ·= K and G/G, a cj/f Aut G G up to isomorph i s m K *Aut C) We now set w(n) = Σ ._ . .. l/*Aut G G up to isoraorphism "^ We can obtain front theorem 3. l' the following properties of the function w(n) : Theorem 3.2t (i) E (ii) Σ (iii) w(<3) = nw(n) 1 w(n)n" w(n) - Πα ( p ||n = ζ (s- H) C (s+2) . . . 30 (This la st formula is due to H all, 1938.) (iv) S n ^ x w (n) - where C - C^ logzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH χ + D M + 0(- i^- *) f (2)ς (3) . . . - 2.294856589... and D^ is an e xp lic it con stan t. I t follows from t h ls theorem t h at there exist po sit ive constants A and B such t h at A/a>(n) £ w (n) £ B/ w(n). Remark » The constant C is well known to be the average number of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA - - - - - 00 ' abelian groups of a given order. However it occurs in that context δs the residue at s - l of the function ς (s) ζ (2s) ς (3s) . . . , which ia quite a different funotion from the function ζ (s+ 1) ς (s+2) . . . . 84 Sample averaqes for imaqinarv quadratic fields It is now a fairly straightforward matter to obtain averages of intereating functions for tmaqinary quadratic fielda. We give here a few aample results (holding of courae only with our heuristlc assumption 1). a) The probability that the odd part of the clasa group is oyclic is ς (2)ζ(3)/(ς (6)zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON Ca 3 <1- φ(1- ^)...) - 97. 757^ approximately. b) This seems very large, but agrees with table counts. If p is an odd prime, the probability that class number is i l - n MI \ ( l- H l- ) . . . - + - - - - — P P p divides the 31 This gives 43.987< for p ·» 3, 23.967^ for p = 5 etc. c) If β 1s a fixed odd Integer, the average number o£ elements of order exactly d) p e in the class group is 1. Write r (G) for the p- rank of an abelian group G. Then if ia an odd prime, the probability that the p- rank of the clasa group be equal to a given integer r is 2 r ~ (ΐ- φ (i- ~i).. ./(1- jjj)2 (i- - ^) 2 ... d - ^ ) 2 . P This decreases very rapidly aa r increases, and helps to explain why no examples of p- rank greater than 5 have been found. r(3t(D}) 2r tfi(D)) e) The average of p p is 2. The average of p p is pzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA + 3. The first of these resulta is particularly signlficant, becausa by a theorem of Davenport and Heilbronn [2], it is known that the average r <J£(D)) of 3 J ia 2. f) If one is interested in resulta concerning nonfundamental dia- crirninanta, one can uae the well known formula h(Df2) where D h(D)f Π | f d - (|)/P) is a negative fundamental discriminant, to obtain heuristic resulta about class numbera. For example the probability that p di- vides the class number ia approximately 52.46644 for p » 3; 24.130< for p =· 5. 85 Real quadratic fields. Since the class groups of real quadratic fields behave quite dif- farently from the class groups of imaginary quadratic fields, it is 32 clearzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA that Λ new heuristic asaumption is necessary. In plain English, it is δs followa ι if we assurae isomorphiam claaaea of groups G weighted by l/#Aut G, our firat heuristic assumption was that the odd part of the class group of imaginary quadratic fields was "randora". F ° r real quadratic fields, we will assume that the odd part of the claas group is of the form G/<g> where g g). is a "random" element in G is "randorn" δs above, and G «g> is the cyclic subgroup generated by With a similar definition of M(f), the assumption is δs followa: Heuristic assumption 2; It is not very eaey to give good Justifications for this assumption. We will give two. Let O The firet one ha a been suggested to us by B. Groas. be the ring of integers of (BU/D), where mental discriminant, and let (p) » β·β. p D ia a negative funda- be a fixed prime which splits in O Then the class group of O [ l/p) ia easily seen to be equal to Μ(τ>'ί/<β>. Now, äs for real quadratic fields the unit rank of 0 [l/p] is equal to l, and a table of such class groups reveals a striking resemblance to tables of claas groups of real quadratic: fields. The second juatification can be cone idered essentially äs due to Gauas, but with deeper insight by D. Shanks [4], It is well known that there is a multipl ication on quadratic forme of equal discriminant called composition. Thie law is not quite a group law on the set of reduced forme since first of all the product of two reduced forms is not reduced, and second the law is not associative. However in a certain sense which we c anno t make precise here it is "almost" a group law 33 [3] [4], On the other hand the aet of reduced forme dividea iteelf under the reduction Operation into cyclea . and the nuraber of auch cy~ cles ia the clasa number. Furtherraore these cyclea can have different number of forme, but they have the aame "length" (eaaentially the regulator) if an appropriate notion of distance ia defined [3], [4]. Finally, in [4] Shanka ahows that the principal cycle, although not a group, diaplays a cyclic-group-like atructure. Thua, although it doea not quite make much sense, it is tempting to Interpret the equation h = hR/R by aaying that the clasa group of a real quadratic field is the quotient of the "group" of reduced forma by the "cyclic aubgroup" formed by the principal cycle. Anyway, if we aaaume heuriatic asaumption 1, eaaentially all the neceaaaey algebraic and analytic results have already been obtained in the imaginary quadratic caee. Thia allowe άB to obtain the following sample reaulta : a) The probability that the odd part of the claaa group is iso- morphic to Tj (where (_| ie a given group of odd order) ia L C M ( 1 — )(1--J. . .) In particular the probability that it ie of order l (t, odd) ie C (l-)<l-)...). 2 2 Thia givea approxiraately 75.446< for i - l, 12.574j( for 1 = 3, 3.772ji for t * 5 etc. b) ia The probability that p dividea the claas nuraber (p odd prima) 34 , Μ,,η^) l c) If (l β of order exactly d) If p P ) (l P 3 )... J,J.- L.- LJ, P P P P P is a flxed odd integer the average number of elements e in the clasa group is 1/e . ie an odd prime, the probability that the p- rank of the clasB group is equal to a given integer r is 1. 1. ., 1.2 _1_.2 l .2. _1 _ (l~- n)<l- ^).../(l- - ) (l—j) ... (1—J) (l- r + 1 P P P P r (H (D)) e) The average of p " is l · * · l/p. The average of 2r (W(D)) , p p is 2 + l/p + l/p . p - r(r- H)M As in the imaginary case, the first of these results is particularly eignificant because by a theorem of Davenport and Heilbronn [2], r (M(D) ) it ie known that the average of 3 is 4/3. 86 Higher degree ftelda. We consider only the case of cyclic extensions of φ degree p. Let Γ ·» <σ> be the Galois group. of prime It ie clear that the class Ρ group ie a Z[r]- module, and even a Ζ[Γ]/(1·*·σ+.. ,+σ ~ )- module, since the norm of an ideal is principal. t o z But this last ring is isomorphic ICpl vihere r is a primitive p- th root of unity, and this is a Dedekind domain, and even a principal ideal domain if p <; 19. It can then be shown that all the theory above can be generalized to this case, the Riemann zeta function being simply replaced by the Dedekind zeta function of the field 01(ζ ). The heuristlc assumption is that the prime to p class group behaves like G/<g>, where G part of the is a "random" finitezyxwvutsrqponmlk Χΐζ module (weighted with l/#Aut G) and <g> is the cyclic Ζ[ζ ]- module 35 generated by a random element in Q. Note that although the unit rank of a typical field is p - l if p ;> 3, äs azyxwvutsrqponmlkjihgfedcbaZYXWV Ζ[ζ ]- module the unit rank is still equal to 1. (If the unit rank ie equal to u it aeems rea- sonable to consider G/<g.,...,g > ) . We give two examples for cvclic cubic fieldsi a) The probability that the prime to 3 part of the class group ie equal to l ia approximately 85.0jpT b) The probability that 4 divides the class number is approxi- mately θ.195<. (Remark! it ia easy to show that if 2 divides the class number, then 4 also divides it.) 87 Concluding remarks. All the above heuristics agree very closely with available tables, and furthermore they agree exactly with the theorems of Davenport and Heilbronn. This seems to give strong Support for the validity of our heuristic assumptions. Extensions of this work in several different directions are under way. Also it would be interesting to know if similar heuristic esti- mates can be made in different contexta, for exrjmple for Tate- Shafarevitch groups of a given elliptic curve twisted by quadratic characters. Finally it is a pleasure to thank our friends and colleagues B. Gross, D, Shanks and L. Washington for interesting discussions on this subject. 38 Bibliography [1] D. Busll, Class groups of guadratic fields, Math. Comp. v. 30, 1976, pp. 610-623. [2] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields II, Proc. Royal. Soc., A, 322 (1971), 405-420. [3] H.W. Lenatra, Jr, , on the calculation of regulators and class nurobers of quadratic fields, pp. 123-150 in J.V. Armitage (ed.), Journees Arithmetiques 1980, London Math. Soc. Lecture Notes Seriee 56, Cambridge university Press, 1982. {4] D. Shanlcs, The infrastructure of a real quadratic field and its applications, Proc. 1972 nurober theory Conference, Boulder, 1972. [5] D. Shanka, H. Williams, in preparation. H. Cohen, L.A. au C.N. R. S. n°226, HER de Mathematigues et Informatique Universite de Bordeaux I 351, Cours de la Liberation 33405 Talence, FRANCE H.W. Lenstra, Jr. Mathematisch Instituut Universiteit van Amsterdam Roetersstraat 15 1018 WB Amsterdam the NETHERLANDS