Quelques résultats effectifs concernant les invariants de Tsfasman-Vlăduţ
Lebacque, Philippe
Annales de l'Institut Fourier, Tome 65 (2015), p. 63-99 / Harvested from Numdam
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On considère dans cet article les propriétés asymptotiques de corps globaux à travers l’étude de leurs invariants de Tsfasman-Vlăduţ, nombres qui décrivent en particulier la décomposition des places dans les tours de corps globaux. On utilise des résultats récents d’Alexander Schmidt et une version faible mais effective du théorème de Grunwald-Wang pour construire des corps globaux infinis ayant un ensemble fini donné d’invariants non nuls et un ensemble prescrit d’invariants nuls, tout en estimant leur défaut.

We consider in this article properties of infinite algebraic extensions of global fields through their Tsfasman-Vladuts invariants, which describe in particular the decomposition of primes in global field towers. We use recent results of A. Schmidt and a weak effective version of the Grunwald-Wang theorem to construct infinite global fields having at the same time a given finite set of positive invariants, a prescribed set of invariants being zero, and a controlled deficiency.

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2925
Classification:  11R29,  11R34,  11R37,  11R45,  11R58
Mots clés: Corps globaux infinis, mild pro-p-groupes, ramification restreinte, théorie du corps de classes 
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     author = {Lebacque, Philippe},
     title = {Quelques r\'esultats effectifs concernant les invariants de Tsfasman-Vl\u adu\c t},
     journal = {Annales de l'Institut Fourier},
     volume = {65},
     year = {2015},
     pages = {63-99},
     doi = {10.5802/aif.2925},
     zbl = {1326.11071},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_2015__65_1_63_0}
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Lebacque, Philippe. Quelques résultats effectifs concernant les invariants de Tsfasman-Vlăduţ. Annales de l'Institut Fourier, Tome 65 (2015) pp. 63-99. doi : 10.5802/aif.2925. http://gdmltest.u-ga.fr/item/AIF_2015__65_1_63_0/

[1] Bach, Eric; Sorenson, Jonathan Explicit bounds for primes in residue classes, Math. Comp., Tome 65 (1996) no. 216, pp. 1717-1735 | Article | MR 1355006 | Zbl 0853.11077

[2] Garcia, Arnaldo; Stichtenoth, Henning Asymptotically good towers of function fields over finite fields, C. R. Acad. Sci. Paris Sér. I Math., Tome 322 (1996) no. 11, pp. 1067-1070 | MR 1396642 | Zbl 0867.11042

[3] Gras, Georges Class field theory, Springer-Verlag, Berlin, Springer Monographs in Mathematics (2003), pp. xiv+491 (From theory to practice, Translated from the French manuscript by Henri Cohen) | Article | MR 1941965 | Zbl 1019.11032

[4] Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers, The Clarendon Press, Oxford University Press, New York (1979), pp. xvi+426 | MR 568909 | Zbl 0058.03301

[5] Ihara, Yasutaka Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Tome 28 (1981) no. 3, p. 721-724 (1982) | MR 656048 | Zbl 0509.14019

[6] Ihara, Yasutaka How many primes decompose completely in an infinite unramified Galois extension of a global field ?, J. Math. Soc. Japan, Tome 35 (1983) no. 4, pp. 693-709 | Article | MR 714470 | Zbl 0518.12006

[7] Kumar Murty, Vijaya; Scherk, John Effective versions of the Chebotarev density theorem for function fields, C. R. Acad. Sci. Paris Sér. I Math., Tome 319 (1994) no. 6, pp. 523-528 | MR 1298275 | Zbl 0822.11077

[8] Labute, John Mild pro-p-groups and Galois groups of p-extensions of ℚ, J. Reine Angew. Math., Tome 596 (2006), pp. 155-182 | Article | MR 2254811 | Zbl 1122.11076

[9] Lagarias, J. C.; Montgomery, H. L.; Odlyzko, A. M. A bound for the least prime ideal in the Chebotarev density theorem, Invent. Math., Tome 54 (1979) no. 3, pp. 271-296 | Article | MR 553223 | Zbl 0401.12014

[10] Lagarias, J. C.; Odlyzko, A. M. Effective versions of the Chebotarev density theorem, Algebraic number fields : L -functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London (1977), pp. 409-464 | MR 447191 | Zbl 0362.12011

[11] Lebacque, P. Sur quelques Propriétés asymptotiques des corps globaux, France, Université de la Méditerranée (2007) (Masters thesis)

[12] Lebacque, Philippe On Tsfasman-Vlăduţ invariants of infinite global fields, Int. J. Number Theory, Tome 6 (2010) no. 6, pp. 1419-1448 | Article | MR 2726590 | Zbl 1225.11146

[13] Louboutin, Stéphane Explicit upper bounds for residues of Dedekind zeta functions and values of L-functions at s=1, and explicit lower bounds for relative class numbers of CM-fields, Canad. J. Math., Tome 53 (2001) no. 6, pp. 1194-1222 | Article | MR 1863848 | Zbl 0998.11066

[14] Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay Cohomology of number fields, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 323 (2008), pp. xvi+825 | Article | MR 2392026 | Zbl 1136.11001

[15] Niederreiter, Harald; Xing, Chaoping Towers of global function fields with asymptotically many rational places and an improvement on the Gilbert-Varshamov bound, Math. Nachr., Tome 195 (1998), pp. 171-186 | Article | MR 1654693 | Zbl 0920.11039

[16] Rosen, Michael Number theory in function fields, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 210 (2002), pp. xii+358 | Article | MR 1876657 | Zbl 0830.11044

[17] Šafarevič, I. R. Extensions with prescribed ramification points, Inst. Hautes Études Sci. Publ. Math. (1963) no. 18, pp. 71-95 | MR 176979 | Zbl 0118.27505

[18] Schmidt, Alexander Über pro-p-fundamentalgruppen markierter arithmetischer kurven, J. Reine Angew. Math., Tome 640 (2010), pp. 203-235 | Article | MR 2629694 | Zbl 1193.14041

[19] Serre, Jean-Pierre Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. (1981) no. 54, pp. 323-401 | Numdam | MR 644559 | Zbl 0496.12011

[20] Serre, Jean-Pierre Rational Points on Curves over Finite Fields (1985) (Harvard University)

[21] Silverman, Joseph H. An inequality relating the regulator and the discriminant of a number field, J. Number Theory, Tome 19 (1984) no. 3, pp. 437-442 | Article | MR 769793 | Zbl 0552.12003

[22] Stichtenoth, Henning Algebraic function fields and codes, Springer-Verlag, Berlin, Universitext (1993), pp. x+260 | MR 1251961 | Zbl 0816.14011

[23] Tsfasman, M. A.; Vlăduţ, S. G. Algebraic-geometric codes, Kluwer Academic Publishers Group, Dordrecht, Mathematics and its Applications (Soviet Series), Tome 58 (1991), pp. xxiv+667 (Translated from the Russian by the authors) | Article | MR 1186841 | Zbl 0727.94007

[24] Tsfasman, M. A.; Vlăduţ, S. G. Infinite global fields and the generalized Brauer-Siegel theorem, Mosc. Math. J., Tome 2 (2002) no. 2, pp. 329-402 (Dedicated to Yuri I. Manin on the occasion of his 65th birthday) | MR 1944510 | Zbl 1004.11037

[25] Tsfasman, Michael; Vlăduţ, Serge; Nogin, Dmitry Algebraic geometric codes : basic notions, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 139 (2007), pp. xx+338 | Article | MR 2339649 | Zbl 1127.94001

[26] Vlèduts, S. G.; DrinfelʼD, V. G. The number of points of an algebraic curve, Funktsional. Anal. i Prilozhen., Tome 17 (1983) no. 1, p. 68-69 | Zbl 0522.14011

[27] Zimmert, Rainer Ideale kleiner Norm in Idealklassen und eine Regulatorabschätzung, Invent. Math., Tome 62 (1981) no. 3, pp. 367-380 | Article | MR 604833 | Zbl 0456.12003