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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.

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  • $\begingroup$ 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

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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:

http://www.cims.nyu.edu/~tschinke/princeton/papers/yuri/jacob/jacob6.pdf

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.

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  • $\begingroup$ 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

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