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12

I am not aware of anybody seriously considering hyperelliptic curves for actual real-world usage, beyond toys, and I would be rather surprised to hear differently from anyone. As you say, hyperelliptic provide comparatively few (if any!) advantages over elliptic curves but have the huge disadvantage that virtually nobody has well-tested, battle-hardened ...


2

The answer is positive: there is a surjective, generically finite morphism $\text{sym}^d(C)\to \text{Prym}(C/C')$, at least away from small characteristics. Fix a $k$-point $x$ of $C$, and use that to define an Abel map, $\alpha_x:C \to J(X)$. The induced composition morphism $$C^g \xrightarrow{\alpha^g} J(C)^g \xrightarrow{\Sigma} J(C),$$ is surjective ...



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