2-Torsion in Jacobians of Curves Over Finite Fields

Let $C$ be a (smooth, projective) curve over a finite field $mathbbF_q$, and let $J_C(mathbbF_q)$ denote its Jacobian. Suppose the genus $g$ of $C$ is at least $1$.

Question 1: Are there curves $C$ for which $J_C(mathbbF_q)$ is isomorphic, as a group, to $(mathbbZ/2mathbbZ)^k$ for some $k$? Can one characterize all such curves?

Question 2: What is known about upper bounds of the size of the $2$-torsion of $J_C(mathbbF_q)$ compared to the size of $J_C(mathbbF_q)$ itself? Is it necessarily true, say, that the 2-torsion is negligible if some parameter ($q$ or $g$) is large enough?

1 Answer
1

I think $y^2=x^9-x$ over $mathbbF_3$ has $J_C(mathbbF_3)$ isomorphic to $(mathbbZ/2)^6$ but please check.

The $2$-torsion in $J_C$ over the algebraic closure is $(mathbbZ/2)^2g$ (or smaller in characteristic two). On the other hand, $#J_C(mathbbF_q) ge (sqrtq -1)^2g$, so for $q > 9$, the latter is bigger than the former (and usually much bigger). So the things you want can only exist for small $q$. I don't know a classification.

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