Nonmaximal primes associated to genus $2$ curve (awaiting review)

The nonmaximal primes associated to a genus $2$ curve are the finitely many primes $\ell$ for which the image of the mod-$\ell$ Galois representation is not as large as possible, subject to constraints imposed by the geometric endomorphism ring of the Jacobian of $C$.

This information is currently available only for genus 2 curves whose Jacobian has geometric endomorphism ring $\Z$, equivalently, curves with Sato-Tate group $\mathrm{USp}(4)$. In this case the nonmaximal primes are simply the primes $\ell$ for which the mod-$\ell$ Galois representation is not surjective.