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A family of abelian varieties rationally isogenous to no Jacobian

Author: James L. Parish
Journal: Proc. Amer. Math. Soc. 106 (1989), 1-7
MSC: Primary 11G10; Secondary 14H40, 14K07
MathSciNet review: 929427
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Abstract: Let $ {E_d}$, for any $ d \in {\mathbf{Q}}{(i)^*}$, be the curve $ {x^3} - dx{z^2} = {y^2}z$, and let $ g$ be any positive integer. It is shown that, if $ d$ is not a square in $ {\mathbf{Q}}(i)$ and $ g > 1$, the abelian variety $ E_d^g$ is not isogenous over $ {\mathbf{Q}}(i)$ to the Jacobian of any genus- $ g$ curve. The proof proceeds by showing that any curve whose Jacobian is isogenous to $ E_d^g$ over $ {\mathbf{Q}}(i)$ must be hyperelliptic, and then showing that no hyperelliptic curve can have Jacobian isogenous to $ E_d^g$ over $ {\mathbf{Q}}(i)$.

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Article copyright: © Copyright 1989 American Mathematical Society