Bielliptic curves and symmetric products

Harris, Joe; Silverman, Joe

doi:10.1090/S0002-9939-1991-1055774-0

Bielliptic curves and symmetric products
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by Joe Harris and Joe Silverman PDF

Proc. Amer. Math. Soc. 112 (1991), 347-356 Request permission

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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