So the question is that, over a finite field, does there exist an abelian variety $A$ for which there does not exist a generically one-to-one morphism from a hyperelliptic curve $C$ to $A$.

p.s. A result of Oort and de Jong proved that a "generic" abelian variety satisfies this property. However it seems generic means outside a countable subfamily in their result.

I'm interested in this question because if we assume all motives over a finite field is semisimple (or just semisimplify it), then by Weil conjecture and Honda-Tate (maybe also Tate's conjecture over finite field), such motive all live in the tensor category of that of abelian varieties, or equivalently that of curves. I am then wondering if there could be any chance that they all actually come from hyperelliptic curves.

  • $\begingroup$ A slight mistake: in the above argument only motives of positive dimensional varieties (but not 0-dim'l ones, they are finite order characters of Galois gp) are generated by curves/abelian varieties. If one might to also allow twist by finite order characters or Tate twist, the question can become more complicated. $\endgroup$ May 16, 2014 at 0:18

1 Answer 1


I don't know the answer, but I'll note that this is explicitly raised as a question by Bogomolov and Tschinkel in remark 8 of this paper:


I'll also remark that a version of this question often comes up among people who study ranks of elliptic curves. For an elliptic curve E/K, is there a hyperelliptic curve X/K in E^r whose Jacobian surjects on E^r? If so, there are infinitely many quadratic twists of E of rank at least r.

  • $\begingroup$ Thanks! I suppose you mean infinitely many quadratic twist of the hyperelliptic curve with rank at least r? $\endgroup$ May 16, 2014 at 13:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.