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Elliptic curves of bounded degree and height
Author:
Joseph H. Silverman
Journal:
Proc. Amer. Math. Soc. 105 (1989), 540-545
MSC:
Primary 11G05; Secondary 14G25, 14K07
MathSciNet review:
953747
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Abstract: We show that there are only finitely many elliptic curves of bounded degree and height, provided that one takes a naive height defined in terms of minimal Weierstrass equations. We show that the corresponding statement is false if instead one uses the Faltings-Parshin modular height.
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Gerd
Faltings, Finiteness theorems for abelian varieties over number
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York, 1986, pp. 9–27. Translated from the German original
[Invent. Math. 73 (1983), no. 3, 349–366; ibid. 75 (1984), no. 2,
381; MR 85g:11026ab] by Edward Shipz. MR
861971
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Hasse, Number theory, Grundlehren der Mathematischen
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Serge
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(85j:11005)
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J. Manin and Y. Zarhin, Heights on families of abelian varieties, Math. USSR-Sb. 18 (1972), 169-179.
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D.
G. Northcott, An inequality in the theory of arithmetic on
algebraic varieties, Proc. Cambridge Philos. Soc. 45
(1949), 502–509. MR 0033094
(11,390a)
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Joseph
H. Silverman, Integer points and the rank of Thue elliptic
curves, Invent. Math. 66 (1982), no. 3,
395–404. MR
662599 (83h:10036), http://dx.doi.org/10.1007/BF01389220
- [7]
Joseph
H. Silverman, Lower bounds for height functions, Duke Math. J.
51 (1984), no. 2, 395–403. MR 747871
(87d:11039), http://dx.doi.org/10.1215/S0012-7094-84-05118-4
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Joseph
H. Silverman, The arithmetic of elliptic curves, Graduate
Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210
(87g:11070)
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Joseph
H. Silverman, Heights and elliptic curves, Arithmetic geometry
(Storrs, Conn., 1984) Springer, New York, 1986, pp. 253–265.
MR
861979
- [1]
- G. Faltings, Finiteness theorems for abelian varieties over number fields, in: Arithmetic Geometry (G. Cornell and J. Silverman, eds.), Springer, New York, 1986. MR 861971
- [2]
- H. Hasse, Number theory, Springer, Berlin, 1969. MR 562104 (81c:12001b)
- [3]
- S. Lang, Fundamentals of diophantine geometry, Springer, New York, 1983. MR 715605 (85j:11005)
- [4]
- J. Manin and Y. Zarhin, Heights on families of abelian varieties, Math. USSR-Sb. 18 (1972), 169-179.
- [5]
- D. G. Northcott, An inequality in the theory of arithmetic on algebraic varieties, Proc. Cambridge Philos. Soc. 45 (1949), 502-518. MR 0033094 (11:390a)
- [6]
- J. H. Silverman, Integer points and the rank of Thue elliptic curves, Invent. Math. 66 (1982), 395-404. MR 662599 (83h:10036)
- [7]
- -, Lower bounds for height functions, Duke Math. J. 51 (1984), 395-403. MR 747871 (87d:11039)
- [8]
- -, The arithmetic of elliptic curves, Springer, New York, 1986. MR 817210 (87g:11070)
- [9]
- -, Heights and elliptic curves, in: Arithmetic Geometry (G. Cornell and J. Silverman, eds.), Springer, New York, 1986. MR 861979
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9939-1989-0953747-8
PII:
S 0002-9939(1989)0953747-8
Article copyright:
© Copyright 1989 American Mathematical Society
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