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Tamely Ramified Towers and Discriminant Bounds for Number Fields

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Compositio Mathematica

Abstract

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R 2m be the minimal root discriminant for totally complex number fields of degree 2m, and put α0 = lim inf m R 2m . One knows that α0 ≥ 4πe γ ≈ 22.3, and, assuming the Generalized Riemann Hypothesis, α0 ≥ 8πe γ ≈ 44.7. It is of great interest to know if the latter bound is sharp. In 1978, Martinet constructed an infinite unramified tower of totally complex number fields with small constant root discriminant, demonstrating that α0 < 92.4. For over twenty years, this estimate has not been improved. We introduce two new ideas for bounding asymptotically minimal root discriminants, namely, (1) we allow tame ramification in the tower, and (2) we allow the fields at the bottom of the tower to have large Galois closure. These new ideas allow us to obtain the better estimate α0 < 83.9.

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References

  1. Ankeny, N.C., Brauer, R. and Chowla, S.: A note on the class numbers of algebraic number fields, Amer. J. Math. 78 (1956), 51-61.

    Google Scholar 

  2. Angles, B. and Maire, C.: A note on tamely ramified towers of global function fields, Preprint, 1999.

  3. Batut, C., Belabas, K., Bernardi, D., Cohen, H. and Olivier, M.: GP/PARI 2.0.7, ftp://megrez.math.u-bordeaux.fr/pub/pari.

  4. Cornell, G. and Rosen, M.: A cohomological investigation of the class Group extension problem, In: P. Ribenboim (ed.), Proc. Queen's Number Theory Conference, Queen's Papers in Pure and Appl. Math. 54, Queen's University Press, 1980, pp. 287-308.

  5. Conway, J. and Sloane, N.: Sphere Packings, Lattices and Groups, Springer, New York, 1988.

    Google Scholar 

  6. Fontaine, J.-M.: Il n'y a pas de variété abelienne sur ℤ, Invent. Math. 81 (1985), 515-538.

    Google Scholar 

  7. Fontaine, J.-M. and Mazur, B.: Geometric Galois representations, In: Elliptic Curves, Modular Forms, and Fermat's Last Theorem (Hong Kong, 1993), Ser. Number Theory I, Internat. Press, Cambridge, MA, 1995, pp. 41-78.

  8. Golod, E. and Shafarevich, I.: On class field towers, Izv. Akad. Nauk SSSR 28 (1964), 261-272 [In Russian]; English transl. in Amer. Math. Soc. Transl. 48. Amer. Math. Soc., Providence, RI, 1965, pp. 91–102.

    Google Scholar 

  9. Hajir, F.: On the growth of p-class groups in p-class field towers, J. Algebra 188 (1997), 256-271.

    Google Scholar 

  10. Hajir, F. and Maire, C.: Unramified subextensions of ray class field towers, Preprint 2001.

  11. Ihara, Y.: How many primes decompose completely in an infinite unramified Galois extension of a global field? J. Math. Soc. Japan 35 (1983), 693-709.

    Google Scholar 

  12. Jehne, W.: On knots in algebraic number theory, J. Reine Angew. Math. 311/312 (1979), 215-254.

    Google Scholar 

  13. Koch, H.: Galoissche Theorie der p-Erweiterungen, VFB Deutscher Verlag der Wissenschaften, Berlin, 1970.

    Google Scholar 

  14. Kondo, T.: Algebraic number fields with the discriminant equal to that of a quadratic number field, J. Math. Soc. Japan 47(1) (1995), 31-36.

    Google Scholar 

  15. Litsyn, S. N. and Tsfasman, M. A.: Constructive high-dimensional sphere packings, Duke Math. J. 54(1) (1987), 147-161.

    Google Scholar 

  16. Maire, C.: Extensions T-ramifiées modérées, S-décomposées, Thesis, Besancon (1995).

  17. Martinet, J.: Tours de corps de classes et estimations de discriminants, Invent. Math. 44 (1978), 65-73.

    Google Scholar 

  18. Martinet, J.: Les tours de corps de classes, unpublished manuscript, 1984.

  19. Martinet, J.: Petits Discriminants, Ann. Inst. Fourier (Grenoble) 29(1) (1979), 159-170.

    Google Scholar 

  20. Masley, J. M.: Odlyzko bounds and class number problems, In: A. Fröhlich (ed.), Algebraic Number Fields (Proc. Durham Symp. 1975), Academic Press, New York, 1977, pp. 465-474

    Google Scholar 

  21. Minkowski, H.: Théorèmes arithmétiques, C. R. Acad. Sci. Paris 112 (1891), 209-212. Reprinted in Ges. Abh. I, pp. 261–263, Chelsea, 1967.

    Google Scholar 

  22. Odlyzko, A. M.: Some analytic estimates of class numbers and discriminants, Invent. Math. 29 (1975), 275-286.

    Google Scholar 

  23. Odlyzko, A. M.: Lower bounds for discriminants of number fields, Acta Arith. 29 (1976), 275-297.

    Google Scholar 

  24. Odlyzko, A. M.: Lower bounds for discriminants of number fields II, Tokoku Math. J. 29 (1977), 209-216.

    Google Scholar 

  25. Odlyzko, A. M.: Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results, Sém. de Théorie des Nombres, Bordeaux 2 (1990), 119-141.

    Google Scholar 

  26. Poitou, G.: Minorations de discriminants (d'après A. M. Odlyzko), Séminaire Bourbaki, Vol. 1975/76, 28ème année, Exp. No. 479, pp. 136-153, Lecture Notes in Math. 567, Springer, New York, 1977.

    Google Scholar 

  27. Roquette, P.: On class field towers, In: J. Cassels and A. Fröhlich (eds), Algebraic Number Theory, Academic Press, New York, 1980.

    Google Scholar 

  28. Schoof, R.: Infinite class field towers of quadratic fields, J. Reine Angew. Math. 372 (1986), 209-220.

    Google Scholar 

  29. Serre, J.-P.: Minorations de discriminants, note of October 1975, In: Collected Papers of Jean-Pierre Serre, Vol. 3, Springer, New York, 1986, pp. 240-243.

    Google Scholar 

  30. Serre, J.-P.: In: Collected Papers of Jean-Pierre Serre, Vol. 3, Springer, New York, 1986, p. 710.

    Google Scholar 

  31. Serre, J.-P.: Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris 296 (1983), ser. I, 397-402. Reprinted in Collected Papers of Jean-Pierre Serre, Vol. 3, Springer, New York, 1986.

    Google Scholar 

  32. Shafarevich, I.: Extensions with prescribed ramification points, Publ. Math. I.H.E.S. 18 (1964), 71-95 [In Russian]; English transl. In: Amer. Math. Soc. Transl. 59. American Math Soc., Providence, RI, 1966, pp. 128–149.

    Google Scholar 

  33. Stark, H. M.: Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135-152.

    Google Scholar 

  34. Tate, J. T.: The non-existence of certain Galois extensions of ℚ unramified outside 2. Arithmetic Geometry (Tempe, AZ, 1993), Contemp. Math. 174, Amer. Math. Soc., Providence, RI, 1994, pp. 153-156.

    Google Scholar 

  35. Tsfasman, M. A. and Vladut, S. G.: Asymptotic properties of global fields and generalized Brauer-Siegel theorem, Prétirage 98–35, Institut Mathématiques de Luminy, 1998.

  36. Weiss, E.: Algebraic Number Theory, Reprint of the 1963 original, Dover Publications, Mineola, NY, 1998.

    Google Scholar 

  37. Yamamura, K.: On infinite unramified Galois extensions of algebraic number fields with many primes decomposing almost completely, J. Math. Soc. Japan 38 (1986), 599-605. Correction, to appear.

    Google Scholar 

  38. Yamamura, K.: Maximal unramified extensions of imaginary quadratic number fields of small conductor, J. Théor. Nombres Bordeaux 9(2) (1997), 405-448.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, UCLA, Los Angeles, CA, 90095, U.S.A.

    Farshid Hajir

  2. Department of Mathematics, California State University, San Marcos, San Marcos, CA, 92096, U.S.A.

    Farshid Hajir

  3. Laboratoire A2X, Université Bordeaux I, Cours de la Libération, 33405, Talence Cedex, France

    Christian Maire

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  1. Farshid Hajir
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Hajir, F., Maire, C. Tamely Ramified Towers and Discriminant Bounds for Number Fields. Compositio Mathematica 128, 35–53 (2001). https://doi.org/10.1023/A:1017537415688

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