I've started reading Hindry-Silverman's Diophantine Geometry and I got a little ahead of myself. Non-isomorphic elliptic curves have non-isomorphic Jacobians, that is because elliptic curves are isomorphic to their Jacobians. However, in the case of two non-isomorphic hyperelliptic curves, are their Jacobians non-isomorphic?
(Non)-Isomorphic Jacobians
3
$\begingroup$
algebraic-geometry
algebraic-curves
abelian-varieties
-
1I'm literally running out the door. Key words to look up: Torelli Theorem. – 2012-08-29
1 Answers
2
As said @Matt in the comments, two (projective smooth... ) curves with isomorphic polarized jacobians are isomorphic.
But if their jacobians are isomorphic just as abstract abelian varieties, then the answer is no. See eg. E. Howe: Infinite families of pairs of curves over Q with isomorphic Jacobians, for a recent account.