Annales mathématiques du Québec

, Volume 37, Issue 1, pp 79–108

# The analogue of the Gauss class number problem in motivic cohomology

• Caroline Junkins
• Manfred Kolster
Original Paper

## Abstract

The classical Gauss problem of determining all imaginary quadratic number fields of class number one has an analogue involving finite motivic cohomology groups attached to the ring of integers $$o_F$$ in a totally real number field $$F$$. In the classical situation, the value of the zeta function of $$F$$ at $$0$$ can be written as a quotient of two—not necessarily coprime—integers, where the numerator is equal to the class number. For a totally real field $$F$$ and an even integer $$n \ge 2$$, the value of the zeta-function of $$F$$ at $$1-n$$ can also be written as a quotient of two—not necessarily coprime—integers, where the numerator is equal to the order $$h_n(F)$$ of the motivic cohomology group $$H_\mathcal M ^2(o_F,\mathbb Z (n))$$. This order is always divisible by $$2^d$$, where $$d$$ is the degree of $$F$$. We determine all totally real fields $$F (\ne \mathbb{Q })$$ and all even integers $$n \ge 2$$, for which the quotient $$\frac{h_n(F)}{2^d}$$ is equal to 1. There are no fields for $$n \ge 6$$, there is only the field $$\mathbb{Q }(\sqrt{5})$$ for $$n = 4$$, and there are 11 fields for $$n=2$$. The motivic cohomology groups $$H_\mathcal M ^2(o_F,\mathbb Z (n))$$ contain the canonical subgroups $$WK^\mathcal M _{2n-2}(F)$$, called motivic wild kernels, which are analogous to Tate–Shafarevic groups. For even integers $$n \ge 4$$, there is again only the case $$n = 4$$ and the field $$\mathbb{Q }(\sqrt{5})$$, for which the motivic wild kernel $$WK^\mathcal M _{2n-2}(F)$$ vanishes. However, for $$n=2$$, among the totally real fields $$F$$ of degrees between $$2$$ and $$9$$, we found 21 of them, for which $$WK^\mathcal M _{2}(F)$$ vanishes, the largest degree being $$5$$. A basic assumption in our approach is the validity of the $$2$$-adic Main Conjecture in Iwasawa theory for the trivial character, which so far has only been proven for abelian number fields (by A. Wiles).

## Keywords

Special values of zeta-functions Gauss problem Motivic cohomology Algebraic $$K$$-theory Iwasawa theory

## Résumé

Un problème classique de Gauss a été de déterminer tous les corps de nombres quadratiques imaginaires ayant un nombre de classes égal à 1. Ce problème a un analogue pour les groupes de cohomologie motivique finis rattachés à l’anneau $$o_F$$ des entiers algébriques d’un corps de nombres $$F$$ totalement réel. Dans le cas classique, les valeurs de la fonction zêta de Dedekind évaluée en 0 peuvent s’écrire comme quotients de deux entiers,  non nécessairement copremiers entre eux, et avec le numérateur égal au nombre de classes. Dans la situation d’un corps $$F$$ totalement réel, avec $$n$$ un entier pair $$\ge 2$$, les valeurs de la fonction zêta de $$F$$ en $$1-n$$ peuvent s’écrire comme quotients de deux entiers, non nécessairement copremiers entre eux, mais avec le numérateur égal à l’ordre $$h_n(F)$$ des groupes de cohomologie motivique $$H_\mathcal M ^2(o_F,{Z}(n))$$. Ces ordres sont toujours divisibles par $$2^d$$, où $$d$$ est le degré de $$F$$. Nous déterminons tous les corps de nombres $$F$$ ($$\ne \mathbb Q$$) totalement réels, et tous les entiers pairs $$n\ge 2$$, pour lesquels $$\frac{h_n(F)}{2^d}=1$$. Il n’y a aucun corps pour $$n\ge 6$$, il n’y a que le corps $$\mathbb Q (\sqrt{5})$$ pour $$n=4$$, et il y a 11 corps totalement réels pour $$n=2$$. Les groupes de cohomologie motivique $$H_\mathcal M ^2(o_F,\mathbb Z (n))$$ contiennent des sous-groupes canoniques $$WK^\mathcal M _{2n-2}(F)$$, appelés noyaux sauvages motiviques, qui sont les analogues des groupes de Tate-Shafarevich. Pour les entiers pairs $$n\ge 4$$, une fois de plus le noyau sauvage motivique $$WK^\mathcal M _{2n-2}(F)$$ s’annule seulement pour le corps $$\mathbb Q (\sqrt{5})$$ et pour $$n=4$$. Cependant, pour $$n=2$$, parmi les corps de nombres totalement réels $$F$$ de degrés entre $$2$$ et $$9$$, nous en avons trouvé 21 pour lesquels $$WK^\mathcal M _{2}(F)$$ s’annnule, le plus grand degré de ces corps étant $$5$$. Une hypothèse de base de notre méthode est la validité de la conjecture principale 2-adique de la théorie d’Iwasawa, qui jusqu’à présent n’a été prouvée que pour les corps de nombres abéliens (par A. Wiles).

## Mathematics Subject Classification

11R23 11R42 11R70 11Y40

## References

1. 1.
Chinburg, T., Kolster, M., Pappas, G., Snaith, V.: Galois structure of $$K$$-groups of rings of integers. $$K$$-Theory 14(4), 319–369 (1998)Google Scholar
2. 2.
Diaz y Diaz, F.: Tables minorant la racine $$n$$-ième du discriminant d’un corps de degré $$n$$. In: Publications Mathématiques d’Orsay 80, vol. 6. Université de Paris-Sud, Département de Mathématiques, Orsay (1980)Google Scholar
3. 3.
Friedlander, E., Suslin, A.: The spectral sequence relating algebraic $$K$$-theory to motivic cohomology. Ann. Sci. École Norm. Sup. (4) 35(6), 773–875 (2002)Google Scholar
4. 4.
Geisser, T.: Motivic cohomology over Dedekind rings. Math. Z. 248(4), 773–794 (2004)
5. 5.
Gillet, H., Soulé, C.: Filtrations on higher algebraic $$K$$-theory. In: Algebraic $$K$$-theory (Seattle, WA, 1997), pp. 89–148. Proc. Sympos. Pure Math., vol. 67. Amer. Math. Soc., Providence (1999)Google Scholar
6. 6.
Goldfeld, D.: Gauss’ class number problem for imaginary quadratic fields. Bull. Am. Math. Soc. 13, 23–37 (1985)
7. 7.
Gross, B., Zagier, D.: Heegner points and derivatives of $$l$$-series. Invent. Math. 84, 225–320 (1986)
8. 8.
Hesselholt, L., Madsen, I.: On the $$K$$-theory of local fields. Ann. Math. (2) 158(1), 1–113 (2003)Google Scholar
9. 9.
Hurrelbrink, J.: On the wild kernel. Arch. Math. 40(4), 316–318 (1983)
10. 10.
Jones, J., Roberts, D.P.: A database of local fields. J. Symb. Comput. 41(1), 80–97 (2006)
11. 11.
Kahn, B.: The Quillen-Lichtenbaum conjecture at the prime 2 (1997, preprint)Google Scholar
12. 12.
Klingen, H.: Über die Werte der Dedekindschen Zetafunktion. Math. Ann. 145, 265–272 (1961)
13. 13.
Kolster, M.: Higher relative class number formulae. Math. Ann. 323(4), 667–692 (2002)
14. 14.
Kolster, M.: $$K$$-theory and arithmetic. In: Contemporary developments in algebraic $$K$$-theory, ICTP Lect. Notes, XV, pp. 191–258 (electronic), Abdus Salam Int. Cent. Theoret. Phys., Trieste (2004)Google Scholar
15. 15.
Kolster, M., Movahhedi, A.: Galois co-descent for étale wild kernels and capitulation. Ann. Inst. Fourier 50(1), 35–65 (2000)
16. 16.
Lichtenbaum, S.: Values of zeta-functions, étale cohomology, and algebraic $$K$$-theory. In: Algebraic $$K$$-theory, II: “Classical” algebraic $$K$$-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 489–501, Lecture Notes in Math., vol. 342. Springer, Berlin (1973)Google Scholar
17. 17.
Martinet, J.: Petits discriminants des corps de nombres. In: Number theory days (1980). (Exeter, 1980), pp. 151–193. London Math. Soc. Lecture Note Ser., vol. 56. Cambridge Univ. Press, Cambridge (1982)Google Scholar
18. 18.
Mazur, B., Wiles, A.: Class fields of abelian extensions of $${ Q}$$. Invent. Math. 76(2), 179–330 (1984)
19. 19.
Mazza, C., Voevodsky, V., Weibel, C.: Lecture notes on motivic cohomology. In: Clay Mathematics Monographs, vol. 2. American Mathematical Society, Providence (2006)Google Scholar
20. 20.
Milne, J.S.: Étale cohomology. In: Princeton Mathematical Series, vol. 33. Princeton University Press, Princeton (1980)Google Scholar
21. 21.
Milnor, J.: Introduction to algebraic $$K$$-theory. In: Annals of Mathematics Studies, vol. 72. Princeton University Press, Princeton (1971)Google Scholar
22. 22.
Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of number fields, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323. Springer, Berlin (2008)Google Scholar
23. 23.
Odlyzko, A.M.: Discriminant bounds (1976, preprint)Google Scholar
24. 24.
Odlyzko, A.M.: Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results. Sém. Théor. Nombres Bordeaux (2) 2(1), 119–141 (1990)Google Scholar
25. 25.
Panin, I.A.: The Hurewicz theorem and $$K$$-theory of complete discrete valuation rings. Math. USSR-Izv. 29(1), 119–131 (1987)
26. 26.
The PARI Group, Bordeaux, PARI/GP, version 2.3.5 (2008)Google Scholar
27. 27.
Poitou, G.: Sur les petits discriminants. In: Séminaire Delange-Pisot-Poitou, 18e année: (1976/77), Théorie des nombres, Fasc. 1 (French), Exp. No. 6, 18 pp., Secrétariat Math., Paris (1977)Google Scholar
28. 28.
Rognes, J., Weibel, C.: Two-primary algebraic $$K$$-theory of rings of integers in number fields. J. Am. Math. Soc. 13(1), 1–54 (2000) (Appendix A by Manfred Kolster)Google Scholar
29. 29.
Serre, J.-P.: Local fields, Translated from the French by Marvin Jay Greenberg, Graduate Texts in Mathematics, vol. 67. Springer, Berlin (1979)Google Scholar
30. 30.
Siegel, C.L.: Berechnung von Zetafunktionen an ganzzahligen Stellen. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1969, 87–102 (1969)Google Scholar
31. 31.
Soulé, C.: $$K$$-théorie des anneaux d’entiers de corps de nombres et cohomologie étale. Invent. Math. 55(3), 251–295 (1979)
32. 32.
Stark, H.M.: A complete determination of the complex quadratic fields of class-number one. Michigan Math. J. 14, 1–27 (1967)
33. 33.
Stein, W., et al.: Sage Mathematics Software (Version 4.4.1), The Sage Development Team (2009)Google Scholar
34. 34.
Suslin, A., Voevodsky, V.: Bloch-Kato conjecture and motivic cohomology with finite coefficients. In: The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), pp. 117–189. NATO Sci. Ser. C Math. Phys. Sci., vol. 548. Kluwer, Dordrecht (2000)Google Scholar
35. 35.
Tate, J.: Symbols in arithmetic. In: Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 201–211. Gauthiers-Villars, Paris (1971)Google Scholar
36. 36.
Tate, J.: Relations between $$K_{2}$$ and Galois cohomology. Invent. Math. 36, 257–274 (1976)
37. 37.
Voevodsky, V.: Triangulated categories of motives over a field, in Cycles, transfers, and motivic homology theories, pp. 188–238. Ann. Math. Stud., vol. 143. Princeton Univ. Press, Princeton (2000)Google Scholar
38. 38.
Voevodsky, V.: On motivic cohomology with $${ Z}/l$$-coefficients. Ann. Math. (2) 174(1), 401–438 (2011)Google Scholar
39. 39.
Voight, J.: Enumeration of totally real number fields of bounded root discriminant. In: Algorithmic number theory, pp. 261–281. Lecture Notes in Comput. Sci., vol. 5011. Springer, Berlin (2008)Google Scholar
40. 40.
Voight, J.: The Gauss higher relative class number problem. Ann. Sci. Math. Québec 32(2), 221–232 (2008)
41. 41.
Wagoner, J.B.: Continuous cohomology and $$p$$-adic $$K$$-theory. In: Algebraic $$K$$-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), pp. 241–248. Lecture Notes in Math., vol. 551. Springer, Berlin (1976)Google Scholar
42. 42.
Washington, L.C.: Introduction to cyclotomic fields, 2nd edn. In: Graduate Texts in Mathematics, vol. 83. Springer, New York (1997)Google Scholar
43. 43.
Watkins, M.: Class numbers of imaginary quadratic fields. Math. Comp. 73(246), 907–938 (2004). (electronic)Google Scholar
44. 44.
Wiles, A.: The Iwasawa conjecture for totally real fields. Ann. Math. (2) 131(3), 493–540 (1990)Google Scholar