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Curves of genus 2 with good reduction away from 2 with a rational Weierstrass point

Published online by Cambridge University Press:  24 October 2008

J. R. Merriman  and
N. P. Smart
J. R. Merriman
Affiliation:
Institute of Mathematics and Statistics, University of Kent, Canterbury
N. P. Smart
Affiliation:
Institute of Mathematics and Statistics, University of Kent, Canterbury
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Abstract

All curves of the title are calculated up to an equivalence relation which is coarser than the relation of isogeny between the associated Jacobian varieties.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 2 , September 1993 , pp. 203 - 214
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Birch, B. J. and Kuyk, W.. Modular Functions of One Variable IV. Springer-Verlag, 1975.Google Scholar
[2]Coghlan, F. B.. Elliptic curves with conductor N = 2m3n. PhD thesis, University of Manchester, 1967.Google Scholar
[3]Cremona, J. E.. Hyperbolic tesselations, modular symbols and elliptic curves over complex quadratic fields. Comp. Math. 51 (1984), 275–323.Google Scholar
[4]Cremona, J. E.. Modular symbols for Γ1(N) and elliptic curves with everywhere good reduction. Math. Proc. Cambridge Philos. Soc. 111 (1992), 199–218.CrossRefGoogle Scholar
[5]Faltings, G.. Endlichkeitssätze für abelsche Varietäten über Zahlenkörpen. Inv. Math. 73 (1983), 349–366.CrossRefGoogle Scholar
[6]Livné, R.. Cubic exponential sums and Galois representations. Contemp. Math. 67 (1987), 247–261.CrossRefGoogle Scholar
[7]Merriman, J. R.. Binary forms and the reduction of curves. D.Phil, thesis, Oxford University, 1970.Google Scholar
[8]Ogg, A. P.. Abelian curves of 2-power conductor. Math. Proc. Cambridge Philos. Soc. 62 (1966), 143–148.CrossRefGoogle Scholar
[9]Oort, F.. Hyperelliptic curves over number fields. In Popp, H. (ed.), Classification of Algebraic Varieties and Compact Complex Manifolds (Springer-Verlag, 1974), 211–218.Google Scholar
[10]Paršin, A. N.. Minimal models of curves of genus 2 and homomorphisms of abelian varieties defined over a field of finite characteristic. Math. of. USSR, Izvestija, 6, (1972), 65–108.CrossRefGoogle Scholar
[11]Pinch, R. G. E.. Elliptic curves with good reduction away from 2. II. Math. Proc. Cambridge Philos. Soc. 100 (1986), 435–457.CrossRefGoogle Scholar
[12]Pinch, R. G. E.. Elliptic curves with good reduction away from 3. Math. Proc. Cambridge Philos. Soc. 101 (1987), 451–459.CrossRefGoogle Scholar
[13]Pohst, M.. On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields. J. Number Theory 14 (1982), 99–117.CrossRefGoogle Scholar
[14]Serre, J. P.. Local Fields. Springer-Verlag, 1979.CrossRefGoogle Scholar
[15]Serre, J. P. and Tate, J. T.. Good reduction of abelian varieties. Ann. Math. 88 (1968), 492–517.Google Scholar
[16]Smart, N. P.. A class of diophantine equations. Publ. Math. Debrecen 41 (1992), 225–229.CrossRefGoogle Scholar
[17]Smart, N. P.. The Computer Solution Of Diophantine Equations. Ph.D. thesis, University of Kent, 1992.Google Scholar
[18]Stroeker, R. J.. Reduction of elliptic curves over imaginary quadratic number fields. Pacific J. Math. 108 (1983), 451–463.CrossRefGoogle Scholar
[19]Top, J.. Hecke L-series related with algebraic cycles or with Siegel modular forms. Ph.D. thesis, Utrecht, 1989.Google Scholar
[20]Šafarevič, I. R.. Algebraic number fields: Proc. Int. Cong. Math. Stockholm. AMS Translations 31 (1963), 25–39.Google Scholar
[21]De Weger, B. M. M.. The weighted sum of two S-units being a square. Indagationes Mathematicae 1 (1990), 243–262.CrossRefGoogle Scholar