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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Bielliptic curves and symmetric products


Authors: Joe Harris and Joe Silverman
Journal: Proc. Amer. Math. Soc. 112 (1991), 347-356
MSC: Primary 11G30; Secondary 14H25
MathSciNet review: 1055774
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Abstract: We show that the twofold symmetric product of a nonhyperelliptic, nonbielliptic curve does not contain any elliptic curves. Applying a theorem of Faltings, we conclude that such a curve defined over a number field $ K$ has only finitely many points over all quadratic extensions of $ K$. We illustrate our theory with the modular curves $ {X_0}(N),{X_1}(N),X(N)$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1055774-0
PII: S 0002-9939(1991)1055774-0
Article copyright: © Copyright 1991 American Mathematical Society