Computing the Hilbert class field of real quadratic fields

Cohen, Henri; Roblot, Xavier-François

doi:10.1090/S0025-5718-99-01111-4

Computing the Hilbert class field of real quadratic fields
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by Henri Cohen and Xavier-François Roblot;

Math. Comp. 69 (2000), 1229-1244

DOI: https://doi.org/10.1090/S0025-5718-99-01111-4

Published electronically: March 10, 1999

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Abstract:

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.

References

Eric Bach and Jonathan Sorenson, Explicit bounds for primes in residue classes, Math. Comp. 65 (1996), no. 216, 1717–1735. MR 1355006, DOI 10.1090/S0025-5718-96-00763-6
C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, User Guide to PARI/GP version 2.0.1, 1997
Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
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). MR 1149802
H. Cohen, F. Diaz y Diaz, M. Olivier, Algorithmic Techniques for Relative Extensions of Number Fields, preprint A2X (1997)
H. Cohen, F. Diaz y Diaz, and M. Olivier, Computing ray class groups, conductors and discriminants, Math. Comp. 67 (1998), no. 222, 773–795. MR 1443117, DOI 10.1090/S0025-5718-98-00912-0
Gary Cornell and Michael Rosen, A note on the splitting of the Hilbert class field, J. Number Theory 28 (1988), no. 2, 152–158. MR 927656, DOI 10.1016/0022-314X(88)90062-5
D. Dummit, B. Tangedal, Computing the Leading Term of an Abelian $L$-function, ANTS III (Buhler Ed.), Lecture Notes in Computer Sci. 1423 (1998), p.400–411
C. Fieker, Computing Class Fields via the Artin Map, preprint, 1997
Eduardo Friedman, Hecke’s integral formula, Séminaire de Théorie des Nombres, 1987–1988 (Talence, 1987–1988) Univ. Bordeaux I, Talence, [1988?], pp. Exp. No. 5, 23. MR 993106
Makoto Ishida, The genus fields of algebraic number fields, Lecture Notes in Mathematics, Vol. 555, Springer-Verlag, Berlin-New York, 1976. MR 435028
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). MR 1484487, DOI 10.1006/jsco.1996.0140
J. Martinet, Character theory and Artin $L$-functions, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London-New York, 1977, pp. 1–87. MR 447187
J. Neukirch, Algebraische Zahlentheorie, Springer-Verlag, Berlin, 1992
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
M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory, Encyclopedia of Mathematics and its Applications, vol. 30, Cambridge University Press, Cambridge, 1989. MR 1033013, DOI 10.1017/CBO9780511661952
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)
Reinhard Schertz, Problèmes de construction en multiplication complexe, Sém. Théor. Nombres Bordeaux (2) 4 (1992), no. 2, 239–262 (French). MR 1208864
R. R. Sharma and Bahman Zohuri, A general method for an accurate evaluation of exponential integrals $E_{1}(x), x>0$, J. Comput. Phys. 25 (1977), no. 2, 199–204. MR 474705, DOI 10.1016/0021-9991(77)90022-5
I.N. Sneddon, The Use of Integral Transforms, Mc Graw-Hill, New York, 1972
H. M. Stark, Values of $L$-functions at $s=1$. I. $L$-functions for quadratic forms, Advances in Math. 7 (1971), 301–343 (1971). MR 289429, DOI 10.1016/S0001-8708(71)80009-9
John Tate, Les conjectures de Stark sur les fonctions $L$ d’Artin en $s=0$, Progress in Mathematics, vol. 47, Birkhäuser Boston, Inc., Boston, MA, 1984 (French). Lecture notes edited by Dominique Bernardi and Norbert Schappacher. MR 782485
Riho Terras, The determination of incomplete gamma functions through analytic integration, J. Comput. Phys. 31 (1979), no. 1, 146–151. MR 531128, DOI 10.1016/0021-9991(79)90066-4
Riho Terras, Generalized exponential operators in the continuation of the confluent hypergeometric functions, J. Comput. Phys. 44 (1981), no. 1, 156–166. MR 642125, DOI 10.1016/0021-9991(81)90042-5