The Chevalley-Herbrand formula and the real abelian main conjecture (new criterion using capitulation of the class group). (English) Zbl 1534.11128
Summary: Analytic theory of real abelian fields \(K\) says (in the cyclic semi-simple case) that the order of the \(p\)-class group \(\mathscr{H}_K\) is equal to the \(p\)-index of cyclotomic units \((\mathscr{E}_K : \mathscr{F}_K)\). We develop, in this article, new promising links between: (i) the Chevalley-Herbrand formula giving the number of “ambiguous classes” in \(p\)-extensions \(L / K\), \(L \subset K(\mu_\ell)\) for auxiliary prime numbers \(\ell \equiv 1 \pmod {2p^N}\) inert in \(K\); (ii) the phenomenon of capitulation of \(\mathscr{H}_K\) in \(L\); (iii) the Main Theorem \(\#\mathscr{H}_{K, \varphi} = (\mathscr{E}_{K, \varphi} : \mathscr{F}_{K, \varphi})\) for the isotypic components using the irreducible \(p\)-adic characters \(\varphi\) of \(K\), that we had conjectured (1977). We prove that this real Main Theorem is trivially fulfilled as soon as \(\mathscr{H}_K\) capitulates in \(L\) (Theorem 1.2). Computations with PARI programs support this new philosophy of the Main Theorem and the very frequent phenomenon of capitulation suggests Conjecture 1.1.
MSC:
| 11R23 | Iwasawa theory |
| 11R20 | Other abelian and metabelian extensions |
| 11R29 | Class numbers, class groups, discriminants |
| 11Y40 | Algebraic number theory computations |
Keywords:
Chevalley-Herbrand formula; real abelian number fields; class groups; main conjecture; \(p\)-adic characters; capitulation of \(p\)-class groups; class field theory; analytic formulasSoftware:
PARI/GPReferences:
| [1] | Bosca, S., Principalization of ideals in abelian extensions of number fields, Int. J. Number Theory, 5, 3, 527-539 (2009) · Zbl 1239.11118 |
| [2] | Chevalley, C., Sur la théorie du corps de classes dans les corps finis et les corps locaux, J. Fac. Sci. Tokyo, 2, 365-476 (1933), Thèse no. 155 · JFM 59.0190.01 |
| [3] | Coates, J.; Sujatha, R., Chap. 6: Main Conjecture, (Cyclotomic Fields and Zeta Values. Cyclotomic Fields and Zeta Values, Springer Monographs in Mathematics (2006), Springer), 89-99 · Zbl 1100.11002 |
| [4] | Ellenberg, J. S.; Pierce, L. B.; Wood, M. M., On ℓ-torsion in class groups of number fields, Algebra Number Theory, 11, 8, 1739-1778 (2017) · Zbl 1398.11136 |
| [5] | Frei, C.; Loughran, D.; Newton, R., Distribution of genus numbers of abelian number fields (2022) |
| [6] | Fukuda, T., Remarks on \(\mathbb{Z}_p\)-extensions of number fields, Proc. Jpn. Acad., Ser. A, 70, 8, 264-266 (1994) · Zbl 0823.11064 |
| [7] | González-Avilés, C. D., Capitulation, ambiguous classes and the cohomology of the units, J. Reine Angew. Math., 613, 75-97 (2007) · Zbl 1226.11122 |
| [8] | González-Avilés, C. D., Chevalley’s ambiguous class number formula for an arbitrary torus, Math. Res. Lett., 15, 6, 1149-1165 (2008) · Zbl 1175.11067 |
| [9] | Gras, G., Étude d’invariants relatifs aux groupes des classes des corps abéliens, Journées Arithmétiques de Caen (1976). Journées Arithmétiques de Caen (1976), Astérisque, 41-42 (1977), 19 pp. · Zbl 0445.12002 |
| [10] | Gras, G., Classes d’idéaux des corps abéliens et nombres de Bernoulli généralisés, Ann. Inst. Fourier, 27, 1, 1-66 (1977) · Zbl 0336.12004 |
| [11] | Gras, G., Nombre de φ-classes invariantes. Application aux classes des corps abéliens, Bull. Soc. Math. Fr., 106, 337-364 (1978) · Zbl 0392.12005 |
| [12] | Gras, G., Principalisation d’idéaux par extensions absolument abéliennes, J. Number Theory, 62, 2, 403-421 (1997) · Zbl 0869.11088 |
| [13] | Gras, G., Class Field Theory: From Theory to Practice, Springer Monographs in Mathematics (2005), Springer, xiii+507 pages |
| [14] | Gras, G., Invariant generalized ideal classes – structure theorems for p-class groups in p-extensions, Proc. Indian Acad. Sci. Math. Sci., 127, 1, 1-34 (2017) · Zbl 1410.11122 |
| [15] | Gras, G., Genus theory and ε-conjectures on p-class groups, J. Number Theory, 207, 423-459 (2020) · Zbl 1436.11134 |
| [16] | Gras, G., Algorithmic complexity of Greenberg’s conjecture, Arch. Math., 117, 277-289 (2021) · Zbl 1469.11420 |
| [17] | Gras, G., Application of the notion of φ-object to the study of p-class groups and p-ramified torsion groups of abelian extensions (2022) |
| [18] | Gras, G., The p-rank ε-conjecture on class groups is true for towers of p-extensions of a number field, J. Ramanujan Math. Soc., 37, 2, 129-138 (2022) · Zbl 1509.11103 |
| [19] | Gras, G., On the λ-stability of p-class groups along cyclic p-towers of a number field, Int. J. Number Theory, 18, 10, 2241-2263 (2022) · Zbl 1520.11094 |
| [20] | Gras, G., Algebraic norm and capitulation of p-class groups in ramified cyclic p-extensions (2022) |
| [21] | Greenberg, R., On the Iwasawa invariants of totally real number fields, Am. J. Math., 98, 1, 263-284 (1976) · Zbl 0334.12013 |
| [22] | Greither, C., Class groups of abelian fields, and the main conjecture, Ann. Inst. Fourier, 42, 3, 449-499 (1992) · Zbl 0729.11053 |
| [23] | Herbrand, J., Nouvelle démonstration et généralisation d’un théorème de Minkowski, C. R. Acad. Sci., 191, 1282-1285 (1930) · JFM 56.0145.02 |
| [24] | Herbrand, J., Sur les classes des corps circulaires, J. Math. Pures Appl. (9), 11, 417-441 (1932) · Zbl 0006.00802 |
| [25] | Herbrand, J., Le développement moderne de la théorie des corps algébriques-Corps de classes et lois de réciprocité. Avec un Appendice par C. Chevalley, (Mémorial des Sciences Mathématiques, Fasc. LXXV (1936), Gauthier-Villars: Gauthier-Villars Paris), 72 pp. · JFM 61.0164.01 |
| [26] | Iwasawa, K., On some modules in the theory of cyclotomic fields, J. Math. Soc. Jpn., 16, 1, 42-82 (1964) · Zbl 0125.29207 |
| [27] | Jaulent, J-F., L’arithmétique des ℓ-extensions (Thèse d’état), Publ. Math. Fac. Sc. Besançon (Algèbre et théorie des nombres), 1, 1, 1-357 (1986) · Zbl 0601.12002 |
| [28] | Jaulent, J-F., L’état actuel du problème de la capitulation, Sém. Théor. Nr. Bordx., 17, 1-33 (1987/1988) · Zbl 0704.11046 |
| [29] | Jaulent, J-F., Classes logarithmiques des corps de nombres, J. Théor. Nr. Bordx., 6, 2, 301-325 (1994) · Zbl 0827.11064 |
| [30] | Jaulent, J-F., Théorie ℓ-adique globale du corps de classes, J. Théor. Nr. Bordx., 10, 2, 355-397 (1998) · Zbl 0938.11052 |
| [31] | Jaulent, J-F., Classes logarithmiques et capitulation, Funct. Approx. Comment. Math., 54, 2, 227-239 (2016) · Zbl 1404.11129 |
| [32] | Jaulent, J-F., Note sur la conjecture de Greenberg, J. Ramanujan Math. Soc., 34, 1, 59-80 (2019) · Zbl 1473.11201 |
| [33] | Jaulent, J-F., Principalisation abélienne des groupes de classes logarithmiques, Funct. Approx. Comment. Math., 61, 2, 257-275 (2019) · Zbl 1450.11120 |
| [34] | Jaulent, J-F., Capitulation abélienne des groupes de classes de rayons, J. Number Theory, 231, 316-332 (2022) · Zbl 1485.11158 |
| [35] | Klüners, J.; Wang, J., ℓ-torsion bounds of the class group of number fields with an ℓ-group as Galois group, Proc. Am. Math. Soc., 150, 2793-2805 (2022) · Zbl 1494.11092 |
| [36] | Kolyvagin, V. A., Euler systems, (Cartier, P.; Katz, N. M.; Manin, Y. I.; Illusie, L.; Laumon, G.; Ribet, K. A., The Grothendieck Festschrift, Vol. II. The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87 (1990), Birkhäuser: Birkhäuser Boston, MA), 435-483 · Zbl 0742.14017 |
| [37] | Koymans, P.; Pagano, C., On the distribution of \(C l(K) [ \ell^\infty]\) for degree ℓ cyclic fields, J. Eur. Math. Soc., 24, 4, 1189-1283 (2022) · Zbl 1534.11117 |
| [38] | Kraft, J. S.; Schoof, R., Computing Iwasawa modules of real quadratic number fields, Compos. Math., 97, 1-2, 135-155 (1995) · Zbl 0840.11043 |
| [39] | Kraft, J. S.; Schoof, R., Erratum: “Computing Iwasawa modules of real quadratic number field” [Compositio Math. 97 (1-2) (1995) 135-155], Compos. Math., 103, 2, 241 (1996) · Zbl 0840.11043 |
| [40] | Kurihara, M., On the ideal class groups of the maximal real subfields of number fields with all roots of unity, J. Eur. Math. Soc., 1, 1, 35-49 (1999) · Zbl 0949.11055 |
| [41] | Lang, S., Cyclotomic Fields I and II, GTM, vol. 121 (1990), Springer-Verlag: Springer-Verlag Berlin, New York, With an appendix by Karl Rubin on a proof of the Main Conjecture (combined 2nd ed.) · Zbl 0704.11038 |
| [42] | Lang, S., Algebraic Number Theory, GTM, vol. 110 (1994), Addison-Wesley Publ. Comp.: Springer-Verlag, corrected third printing, 2000 · Zbl 0811.11001 |
| [43] | Leopoldt, H. W., Über Einheitengruppe und Klassenzahl reeller abelscher Zahlkörper, Abh. Deutsche Akad. Wiss. Berlin, Math., 2 (1953), 47 pp. |
| [44] | Leopoldt, H. W., Zur Arithmetik in abelschen Zahlkörpern, J. Reine Angew. Math., 209, 54-71 (1962) · Zbl 0204.07101 |
| [45] | Li, J.; Yu, C. F., The Chevalley-Gras formula over global fields, J. Théor. Nr. Bordx., 32, 2, 525-543 (2020) · Zbl 1465.11217 |
| [46] | Li, J.; Ouyang, Y.; Xu, Y.; Zhang, S., ℓ-class groups of fields in Kummer towers, Publ. Mat., 66, 1, 235-267 (2022) · Zbl 1490.11108 |
| [47] | Maire, C., \(T - S\) capitulation, Publ. Math. Fac. Sc. Besançon (Algèbre et théorie des nombres), Article 2 pp. (1996) |
| [48] | Maire, C., Une remarque sur la capitulation du groupe des classes au sens restreint, Publ. Math. Fac. Sc. Besançon (Algèbre et théorie des nombres), Article 2 pp. (1998) |
| [49] | Maire, C., Genus theory and governing fields, N.Y. J. Math., 24, 1056-1067 (2018) · Zbl 1442.11155 |
| [50] | Mizusawa, Y.; Yamamoto, K., On p-class groups of relative cyclic p-extensions, Arch. Math., 117, 3, 253-260 (2021) · Zbl 1480.11145 |
| [51] | Pagani, L., Greenberg’s conjecture for real quadratic fields and the cyclotomic \(\mathbb{Z}_2\)-extension, Math. Comput., 91, 1437-1467 (2022) · Zbl 1522.11114 |
| [52] | The PARI Group, PARI/GP, version 2.5.3, Université de Bordeaux |
| [53] | Perrin-Riou, B., Travaux de Kolyvagin et Rubin, Séminaire Bourbaki: Tome 32 (1989/90), Exposés 715-729. Séminaire Bourbaki: Tome 32 (1989/90), Exposés 715-729, Astérisque, 189-190, 717 (1990), 38 pp. |
| [54] | Pierce, L. B., Counting problems: class groups, primes, and number fields, (Proceedings of the ICM 2022 (2022)) |
| [55] | Pierce, L. B.; Turnage-Butterbaugh, C. L.; Wood, M. M., An effective Chebotarev density theorem for families of number fields, with an application to ℓ-torsion in class groups, Invent. Math., 219, 2, 701-778 (2020) · Zbl 1445.11129 |
| [56] | Ribet, K. A., A modular construction of unramified p-extensions of \(\mathbb{Q}( \mu_p)\), Invent. Math., 34, 151-162 (1976) · Zbl 0338.12003 |
| [57] | Ribet, K. A., Bernoulli numbers and ideal classes, Gazette SMF, 118, 42-49 (2008) · Zbl 1196.11145 |
| [58] | Rubin, K., The Main Conjecture, GTM, vol. 121, 397-419 (1990), Springer-Verlag, Appendix to Cyclotomic Fields I & II by S. Lang · Zbl 0704.11038 |
| [59] | Serre, J.-P., Représentations linéaires des groupes finis, cinquième édition corrigée et augmentée (1998), Coll. Méthodes, Hermann · Zbl 0926.20003 |
| [60] | Smith, A., The distribution of \(\ell^\infty \)-Selmer groups in degree ℓ twist families (2022) |
| [61] | Solomon, D., On the class groups of imaginary abelian fields, Ann. Inst. Fourier, 40, 3, 467-492 (1990) · Zbl 0694.12004 |
| [62] | Thaine, F., On the ideal class groups of real abelian number fields, Ann. Math. (2), 128, 1, 1-18 (1988) · Zbl 0665.12003 |
| [63] | Wang, J., Pointwise bound for ℓ-torsion in class groups: elementary abelian extensions, J. Reine Angew. Math., 2021, 773, 129-151 (2020) · Zbl 1468.11227 |
| [64] | Washington, L. C., Introduction to Cyclotomic Fields, GTM, vol. 83 (1997), Springer · Zbl 0966.11047 |
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