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For which (g,q) does there exist a supersingular curve?

We say a curve over a finite field $\mathbb F_q$ is supersingular if it's Frobenius eigenvalues (on $H^1(X,\mathbb Z_\ell)$) are of the form $q^{1/2}\alpha$ for $\alpha$ a root of unity. As far as I ...
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