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Elliptic curves with good reduction away from 3

Published online by Cambridge University Press:  24 October 2008

R. G. E. Pinch
R. G. E. Pinch
Affiliation:
Emmanuel College, Cambridge
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Extract

In this paper we list the elliptic curves defined over Q(√− 3) with good reduction away from the prime dividing 3. As in [8] and [9] a discriminant estimate is used to show that such a curve must have a subgroup of order 3 defined over Q(√−3).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 3 , May 1987 , pp. 451 - 459
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1]Birch, B. J. and Kuyk, W. (eds.). Modular Functions of One Variable: IV. Lecture Notes in Math. vol. 476 (Springer, 1975).CrossRefGoogle Scholar
[2]Coghlan, F. B.. Elliptic curves with conductor N = 2m3n. Ph.D. thesis, University of Manchester 1967.Google Scholar
[3]Cremona, J. E.. Hyperbolic tesselations, modular symbols and elliptic curves over complex quadratic fields. Compositio Math. 51 (1984), 275–323.Google Scholar
[4]Fricke, R.. Lehrbuch der Algebra (in 3 vols) (Vieweg, 1924, 1926, 1928).Google Scholar
[5]Hasse, H.. Number Theory. Grundlehren Math. Wiss. 229 (Springer, 1980).CrossRefGoogle Scholar
[6]Klein, F. and Fricke, R.. Vorlesungen über die Theorie der elliptischen Modulfunktionen (Teubner, 1966).Google Scholar
[7]Pinch, R. G. E.. Elliptic curves over number fields. D.Phil, thesis, University of Oxford 1982.Google Scholar
[8]Pinch, R. G. E.. Elliptic curves with good reduction away from 2. Math. Proc. Cambridge Philos. Soc. 96 (1984), 25–38.CrossRefGoogle Scholar
[9]Pinch, R. G. E.. Elliptic curves with good reduction away from 2: II. Math. Proc. Cambridge Philos. Soc. 100 (1986), 435–457.CrossRefGoogle Scholar
[10]Serre, J.-P. and Tate, J.. Good reduction of Abelian varieties. Ann. Math. 88 (1968), 492–527.CrossRefGoogle Scholar
[11]Silverman, J. H.. The Arithmetic of Elliptic Curves. Graduate Texts in Math. 106 (Springer, 1986).CrossRefGoogle Scholar
[12]Stroeker, R. J.. Elliptic curves defined over imaginary quadratic number fields. Ph.D. thesis, University of Amsterdam 1975.Google Scholar
[13]Stroeker, R. J.. Reduction of elliptic curves over imaginary quadratic number fields. Pacific J. Math. 108 (1983), 451–463.CrossRefGoogle Scholar
[14]Tate, J.. Algorithm for defining the type of a singular fibre in an elliptic pencil. In [1], 33–52.CrossRefGoogle Scholar
[15]Tate, J.. The arithmetic of elliptic curves. Invent. Math. 23 (1974), 33–52.CrossRefGoogle Scholar
[16]Val, P. du. Elliptic functions and elliptic curves. London Math. Society Lecture Notes Series no. 9 (Cambridge University Press, 1973).CrossRefGoogle Scholar
[17]Washington, L. C.. Introduction to cydotomic fields. Graduate Texts in Math. 83 (Springer, 1982).CrossRefGoogle Scholar
[18]Weiss, E.. Algebraic Number Theory (Chelsea, 1963).Google Scholar