Abstract
The study of the maximal p -extension of a global field k unramified everywhere and totally split at a finite set of places of k has at least two important applications: it gives information on the asymptotic behavior of discriminants versus degree in the number field case (as measured by the Martinet constant a(t)), and on the relationship between genus and the number of places of degree one (for large genus) in the function field case (as measured by the Ihara constant A(q)). We survey recent work on class-fieldtheoretical constructions of towers of global fields which are optimal for the study of these phenomena, including best known examples in both settings; these contain, among others, an infinite unramified tower of totally complex number fields with small root discriminant improving Martinet’s record. We show that allowing wild ramification to limited depth leads to asymptotically good towers. However, we demonstrate also that the investigation of the infinitude of these towers involves difficulties absent in the tame case.
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References
W. Aitken and F. Hajir, Some asymptotically good towers of function fields, in preparation.
B. Angles and C. Maire, A note on tamely ramified towers of global functions fields, Preprint, 1999.
V. G. Drinfeld and S. G. Vladut, Number of points of an algebraic curve, Funct. Anal 17 (1983), 53–54.
N. Elkies, Explicit Modular Towers, Proceedings of the Thirty-Fifth Annual Aller-ton Conference on Communication, Control and Computing (1997, T. Basar and A. Vardy, eds), Univ. of Illinois at Urbana-Champaign 1998, pp. 23–32.
N. Elkies, A. Kresch, B. Poonen, J. Wetherell and M. Zieve, Curves of every genus with many points II: asymptotically good families, in preparation.
G. Frey, E. Kani and H. Völklein, Curves with infinite K-rational geometric fundamental group in Aspects of Galois Theory (Gainesville, FL, 1996), 85–118, London Math. Soc. Lecture Note Ser., 256, Cambridge Univ. Press, Cambridge, 1999, 46 (1994), 467–476.
A. Garcia and H. Stichtenoch, A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound, Invent. Math. 121 (1995), 211–222.
N. L. Gordeev, Infinitude of the number of relations in the Galois group of the maximal p-extension of a local field with restricted ramification, Math. USSR 18 (1982), 513–524.
K. Haberland, Galois cohomology of algebraic number fields, V.E.B. Deutscher Verlag der Wissenschaften, 1970.
F. Hajir and C. Maire, Tamely ramified towers and discriminants bounds for number fields,Comp. Math., to appear.
F. Hajir and C. Maire, Tamely ramified towers and discriminants bounds for number fields II, Preprint, 2000.
F. Hajir and C. Maire, Extensions of number fields with wild ramification of bounded depth, Preprint, 2000.
D. R. Hayes, Explicit class field theory for rational function fields, Trans. Amer. Math. Soc. 189 (1974), 77–91.
Y Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Tokyo 28 (1981), 721–724.
H. Koch, Galoissche Theorie der p-Erweiterungen, VFB Deutscher Verlag der Wissenschaften, Berlin, 1970.
T. Kondo, Algebraic number fields with the discriminant equal to that of quadratic number field, J. Math. Soc. Japan 47 (1995), 31–36.
W. W. Li and H. Maharaj, Coverings of curves with asymptotically many rational points, Preprint, 1999.
J. Martinet, Tours de corps de classes et estimations de discriminants, Invent. Math. 44 (1978), 65–73.
H. Niederreiter and C. Xing, Towers of global function fields with asymptotically many rational places and an improvement on the Gilbert-Varshamov bound, Math. Nach 195 (1998), 171–186.
H. Niederreiter and C. Xing, A counterexample to Perret’s conjecture on infinite class field towers for global function fields, Finite Fields Appl. 5 (1999), 240–245.
H. Niederreiter and C. Xing, Curve sequences with asymptotically many rational points in Applications of Curves Over Finite Fields (ed. M. Fried) Contemp. Math. vol. 245, American Math. Soc., 1999.
A. M. Odlyzko, Lower bounds for discriminants of number fields II, Tokoku Math. J. 29 (1977), 209–216.
]A. M. Odlyzko, 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.
M. Perret, Tours ramifiées infinies de corps de classes, J Number Theory 38 (1991), 300–322.
G. Poitou, Minorations de discriminants (d’aprés A. M. Odlyzko), Séminaire Bourbaki, Vol. 1975/76, 28ème année, Exp. No. 479, pp. 136–153, Lecture Notes in Math. 567, Springer-Verlag, 1977.
P. Roquette, On Class field towers, in Algebraic Number Theory, ed. J. Cassels, A. Fröhlich, Academic Press, 1980.
R. Schoof, Algebraic curves over IF2 with many rational points, J. Number Theory 41 (1992), 6–14.
R. Schoof, Infinite class field towers of quadratics fields, J. Reine Angew. Math. 372 (1986), 209–220.
J.-P. Serre, Corps locaux, Hermann, Paris, 1962.
J.-P. Serre, Minorations de discriminants, note of October 1975, published on pp. 240–243 in vol. 3 of Jean-Pierre Serre, Collected Papers, Springer-Verlag, 1986.
J.-P. Serre, Rational Points on Curves Over Finite Fields, Harvard Course Notes by F. Gouvea (unpublished), 1985.
H. M. Stark, Some effetive cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135–152.
A. Temkine, Hilbert class field towers of function fields over finite fields and lower bounds for A(q), Preprint, 1999.
M. A. Tsfasman and S. G. Vladut, Asymptotic properties of global fields and generalized Brauer-Siegel Theorem, Prétirage 98–35, Institut Mathématiques de Luminy, 1998.
M. A. Tsfasman, S. G. Vladut and T. Zink, Modular curves, Shimura curves and Goppa codes better than the Varshamov-Gilbert bound,Math. Nach. 109 (1982), 21–28.
K. Wingberg, Galois groups of local and global type, J. reine angew. Math. 517 (1999), 223–239.
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Hajir, F., Maire, C. (2001). Asymptotically Good Towers of Global Fields. In: Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S. (eds) European Congress of Mathematics. Progress in Mathematics, vol 202. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8266-8_16
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DOI: https://doi.org/10.1007/978-3-0348-8266-8_16
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