Ideal class groups, Hilbert’s irreducibility theorem, and integral points of bounded degree on curves
Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 485-499.

Nous étudions la construction et le comptage, pour tout couple d’entiers m,n>1, des corps de nombres de degré n dont le groupe des classes possède un “grand” m-rang. Notre technique repose essentiellement sur le théorème d’irréductibilité de Hilbert et sur des résultats concernant les points entiers de degré borné sur des courbes.

We study the problem of constructing and enumerating, for any integers m,n>1, number fields of degree n whose ideal class groups have “large" m-rank. Our technique relies fundamentally on Hilbert’s irreducibility theorem and results on integral points of bounded degree on curves.

DOI : 10.5802/jtnb.598
Levin, Aaron 1

1 Centro di Ricerca Matematica Ennio De Giorgi Collegio Puteano Scuola Normale Superiore Piazza dei Cavalieri, 3 I-56100 Pisa, Italy
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Levin, Aaron. Ideal class groups, Hilbert’s irreducibility theorem, and integral points of bounded degree on curves. Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 485-499. doi : 10.5802/jtnb.598. http://www.numdam.org/articles/10.5802/jtnb.598/

[1] N. C. Ankeny and S. Chowla, On the divisibility of the class number of quadratic fields. Pacific J. Math. 5 (1955), no. 3, 321–324. | MR | Zbl

[2] T. Azuhata and H. Ichimura, On the divisibility problem of the class numbers of algebraic number fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 579–585. | MR | Zbl

[3] F. Beukers and S. Tengely, An implementation of Runge’s method for Diophantine equations. Preprint.

[4] Y. F. Bilu and F. Luca, Divisibility of class numbers: enumerative approach. J. Reine Angew. Math. 578 (2005), 79–91. | MR | Zbl

[5] E. Bombieri, On Weil’s “théorème de décomposition”. Amer. J. Math. 105 (1983), no. 2, 295–308. | Zbl

[6] A. Brumer, Ramification and class towers of number fields. Michigan Math. J. 12 (1965), no. 2, 129–131. | MR | Zbl

[7] A. Brumer and M. Rosen, Class number and ramification in number fields. Nagoya Math. J. 23 (1963), 97–101. | MR | Zbl

[8] K. Chakraborty and R. Murty, On the number of real quadratic fields with class number divisible by 3. Proc. Amer. Math. Soc. 131 (2003), no. 1, 41–44 (electronic). | MR | Zbl

[9] S. D. Cohen, The distribution of Galois groups and Hilbert’s irreducibility theorem. Proc. London Math. Soc. (3) 43 (1981), no. 2, 227–250. | Zbl

[10] R. Dvornicich and U. Zannier, Fields containing values of algebraic functions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), no. 3, 421–443. | Numdam | MR | Zbl

[11] R. Dvornicich and U. Zannier, Fields containing values of algebraic functions. II. (On a conjecture of Schinzel). Acta Arith. 72 (1995), no. 3, 201–210. | MR | Zbl

[12] S. Hernández and F. Luca, Divisibility of exponents of class groups of pure cubic number fields. High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Fields Inst. Commun., vol. 41, Amer. Math. Soc., Providence, RI, 2004, pp. 237–244. | MR | Zbl

[13] P. Humbert, Sur les nombres de classes de certains corps quadratiques. Comment. Math. Helv. 12 (1940), no. 1, 233–245. | MR | Zbl

[14] H. Ichimura, On 2-rank of the ideal class groups of totally real number fields. Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 7, 329–332. | MR | Zbl

[15] M. Ishida, On 2-rank of the ideal class groups of algebraic number fields. J. Reine Angew. Math. 273 (1975), 165–169. | MR | Zbl

[16] S. Kuroda, On the class number of imaginary quadratic number fields. Proc. Japan Acad. 40 (1964), 365–367. | MR | Zbl

[17] A. Levin, Vojta’s inequality and rational and integral points of bounded degree on curves. Compos. Math. 143 (2007), no. 1, 73–81.

[18] F. Luca, A note on the divisibility of class numbers of real quadratic fields. C. R. Math. Acad. Sci. Soc. R. Can. 25 (2003), no. 3, 71–75. | MR | Zbl

[19] R. Murty, Exponents of class groups of quadratic fields. Topics in number theory (University Park, PA, 1997), Math. Appl., vol. 467, Kluwer Acad. Publ., Dordrecht, 1999, pp. 229–239. | MR | Zbl

[20] T. Nagell, Über die Klassenzahl imaginär-quadratischer Zahlkörper. Abh. Math. Sem. Univ. Hamburg 1 (1922), 140–150.

[21] T. Nagell, Collected papers of Trygve Nagell. Vol. 1. Queen’s Papers in Pure and Applied Mathematics, vol. 121, Queen’s University, Kingston, ON, 2002. | Zbl

[22] S. Nakano, On the construction of certain number fields. Tokyo J. Math. 6 (1983), no. 2, 389–395. | MR | Zbl

[23] S. Nakano, On ideal class groups of algebraic number fields. J. Reine Angew. Math. 358 (1985), 61–75. | MR | Zbl

[24] P. Roquette and H. Zassenhaus, A class of rank estimate for algebraic number fields. J. London Math. Soc. 44 (1969), 31–38. | MR | Zbl

[25] J.-P. Serre, Lectures on the Mordell-Weil theorem, third ed. Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1997. | MR | Zbl

[26] K. Soundararajan, Divisibility of class numbers of imaginary quadratic fields. J. London Math. Soc. (2) 61 (2000), no. 3, 681–690. | MR | Zbl

[27] V. G. Sprindžuk, Reducibility of polynomials and rational points on algebraic curves. Dokl. Akad. Nauk SSSR 250 (1980), no. 6, 1327–1330. | MR | Zbl

[28] K. Uchida, Class numbers of cubic cyclic fields. J. Math. Soc. Japan 26 (1974), no. 3, 447–453. | MR | Zbl

[29] P. Vojta, A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing. J. Amer. Math. Soc. 5 (1992), no. 4, 763–804. | MR | Zbl

[30] P. J. Weinberger, Real quadratic fields with class numbers divisible by n. J. Number Theory 5 (1973), no. 3, 237–241. | MR | Zbl

[31] Y. Yamamoto, On unramified Galois extensions of quadratic number fields. Osaka J. Math. 7 (1970), 57–76. | MR | Zbl

[32] G. Yu, A note on the divisibility of class numbers of real quadratic fields. J. Number Theory 97 (2002), no. 1, 35–44. | MR | Zbl

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