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Computing the Hilbert class field of real quadratic fields. (English) Zbl 1042.11075

Summary: Using the units appearing in Stark’s conjectures on the values of \(L\)-functions at \(s=0\), we give a complete algorithm for computing an explicit generator of the Hilbert class field of a real quadratic field.

MSC:

11Y40 Algebraic number theory computations
11R37 Class field theory
11R42 Zeta functions and \(L\)-functions of number fields
11Y35 Analytic computations

Software:

PARI/GP
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References:

[1] Eric Bach and Jonathan Sorenson, Explicit bounds for primes in residue classes, Math. Comp. 65 (1996), no. 216, 1717 – 1735. · Zbl 0853.11077
[2] C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, User Guide to PARI/GP version 2.0.1, 1997
[3] Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. · Zbl 0786.11071
[4] Henri Cohen and Francisco Diaz y Diaz, A polynomial reduction algorithm, Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 2, 351 – 360 (English, with French summary). · Zbl 0758.11053
[5] H. Cohen, F. Diaz y Diaz, M. Olivier, Algorithmic Techniques for Relative Extensions of Number Fields, preprint A2X (1997)
[6] H. Cohen, F. Diaz y Diaz, and M. Olivier, Computing ray class groups, conductors and discriminants, Math. Comp. 67 (1998), no. 222, 773 – 795. · Zbl 0929.11064
[7] Gary Cornell and Michael Rosen, A note on the splitting of the Hilbert class field, J. Number Theory 28 (1988), no. 2, 152 – 158. · Zbl 0633.12003
[8] D. Dummit, B. Tangedal, Computing the Leading Term of an Abelian \(L\)-function, ANTS III , Lecture Notes in Computer Sci. 1423 (1998), p.400-411 · Zbl 0918.11059
[9] C. Fieker, Computing Class Fields via the Artin Map, preprint, 1997 · Zbl 0982.11074
[10] Eduardo Friedman, Hecke’s integral formula, Séminaire de Théorie des Nombres, 1987 – 1988 (Talence, 1987 – 1988) Univ. Bordeaux I, Talence, 198?, pp. Exp. No. 5, 23. · Zbl 0697.12010
[11] Makoto Ishida, The genus fields of algebraic number fields, Lecture Notes in Mathematics, Vol. 555, Springer-Verlag, Berlin-New York, 1976. · Zbl 0353.12001
[12] Jürgen Klüners and Michael Pohst, On computing subfields, J. Symbolic Comput. 24 (1997), no. 3-4, 385 – 397. Computational algebra and number theory (London, 1993). · Zbl 0886.11072
[13] J. Martinet, Character theory and Artin \?-functions, Algebraic number fields: \?-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 1 – 87.
[14] J. Neukirch, Algebraische Zahlentheorie, Springer-Verlag, Berlin, 1992 · Zbl 0747.11001
[15] M. Daberkow, M. Pohst, Computations with Relative Extensions of Number Fields with an Application to the Construction of Hilbert Class Fields, Proc. ISAAC’95, ACM Press, New-York 1995, 68-76 · Zbl 0930.11089
[16] M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory, Encyclopedia of Mathematics and its Applications, vol. 30, Cambridge University Press, Cambridge, 1989. · Zbl 0685.12001
[17] X.-F. Roblot, Stark’s Conjectures and Hilbert’s Twelfth Problem, preprint; Algorithmes de Factorisation dans les Extensions Relatives et Applications de la Conjecture de Stark à la Construction des Corps de Classes de Rayon, Thesis, Université Bordeaux I (1997)
[18] Reinhard Schertz, Problèmes de construction en multiplication complexe, Sém. Théor. Nombres Bordeaux (2) 4 (1992), no. 2, 239 – 262 (French). · Zbl 0797.11083
[19] R. R. Sharma and Bahman Zohuri, A general method for an accurate evaluation of exponential integrals \?\(_{1}\)(\?),\?>0, J. Computational Phys. 25 (1977), no. 2, 199 – 204. · Zbl 0401.65009
[20] I.N. Sneddon, The Use of Integral Transforms, Mc Graw-Hill, New York, 1972 · Zbl 0237.44001
[21] H. M. Stark, Values of \?-functions at \?=1. I. \?-functions for quadratic forms., Advances in Math. 7 (1971), 301 – 343 (1971). , https://doi.org/10.1016/S0001-8708(71)80009-9 H. M. Stark, \?-functions at \?=1. II. Artin \?-functions with rational characters, Advances in Math. 17 (1975), no. 1, 60 – 92. , https://doi.org/10.1016/0001-8708(75)90087-0 H. M. Stark, \?-functions at \?=1. III. Totally real fields and Hilbert’s twelfth problem, Advances in Math. 22 (1976), no. 1, 64 – 84. , https://doi.org/10.1016/0001-8708(76)90138-9 Harold M. Stark, \?-functions at \?=1. IV. First derivatives at \?=0, Adv. in Math. 35 (1980), no. 3, 197 – 235. · Zbl 0475.12018
[22] John Tate, Les conjectures de Stark sur les fonctions \? d’Artin en \?=0, Progress in Mathematics, vol. 47, Birkhäuser Boston, Inc., Boston, MA, 1984 (French). Lecture notes edited by Dominique Bernardi and Norbert Schappacher. · Zbl 0545.12009
[23] Riho Terras, The determination of incomplete gamma functions through analytic integration, J. Comput. Phys. 31 (1979), no. 1, 146 – 151. · Zbl 0412.33002
[24] Riho Terras, Generalized exponential operators in the continuation of the confluent hypergeometric functions, J. Comput. Phys. 44 (1981), no. 1, 156 – 166. · Zbl 0474.65012
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