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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 know, the question: "For which pairs $(g,q)$ does there exist a supersingular curve of genus $g$ over $\mathbb F_q$" is open. Partial results that I know are:

  1. Over characteristic $2$, every genus occurs: van der Geer, van der Vlugt.
  2. Genus $4$ in arbitrary characteristic is known: https://arxiv.org/abs/1903.08095
  3. The Hermitian curve $y^q + y = x^{q+1}$ is supersingular for $q = p^n$. slide 20.
  4. The Supersingularity of Hurwitz Curves proves for a specific farmily of $(g,q)$.
  5. Theorem 1.1 here lists, for $4\leq g\leq 11$, some congruence conditions on $p$ for which there exist supersingular curves.

I am sure I have missed a few but I are there are any large families that I have missed? What is the state of the art today?

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