Abstract
The abelian extensions of an imaginary quadratic field can theoretically be generated by the values of the modular j-function, but these values are too large to be useful in practice. We show how Shimura's reciprocity law can be applied to find small generators for these extensions, and to compute the corresponding irreducible polynomials.
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References
Birch B.: Weber's class invariants. Mathematika 16 (1969) 283–294
Cox D.A.: Primes of the form x 2 + ny 2. Wiley-Interscience (1989)
Gee, A.C.P.: thesis Universiteit van Amsterdam (in preparation)
Lang, S.: Elliptic functions, 2nd edition. Springer Graduate Text in Mathematics 112 (1987)
Schertz, R.: Die singulÄren Werte der Weberschen Funktionen f, f1, f2,γ2,γ3. J. Reine Angew. Math. 286/287 (1976) 46–74
Schertz, R.: Problèmes de construction en multiplication complexe. Sém. Théor. Nombres Bordeaux, (2) 4 (1992) no. 2, 239–262
Shimura G.: Introduction to the Arithmetic Theory of Automorphic Functions. Iwanami Shoten and Princeton University Press (1971)
Shimura, G.: Complex Multiplication. Modular functions of One Variable I Springer Lecture Notes in Mathematics 320 (1973)
Stark, H.M. On a ”gap” in a theorem of Heegner. J. Number Theory 1 (1969) 16–27
Washington, L.C.: Introduction to Cyclotomic Fields, 2nd edition. Springer Graduate Text in Mathematics 83 (1997)
Weber, H.: Lehrbuch der Algebra, Vol. III. Chelsea reprint original edition 1908
Yui, N. and Zagier, D.: On the singular values of Weber modular functions. Math. Comp. 66 (1997) no. 20, 1645–1662
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© 1998 Springer-Verlag Berlin Heidelberg
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Gee, A., Stevenhagen, P. (1998). Generating class fields using Shimura reciprocity. In: Buhler, J.P. (eds) Algorithmic Number Theory. ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054883
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DOI: https://doi.org/10.1007/BFb0054883
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