Firstly, applying the unitary matrix heuristic to deduce tiny trace would reasonably require that the Galois action on unstable cohomology is close to irreducible. Is that plausible? There may be natural “Hecke-type” correspondences between different M_gs, like “curve X is a covering of curve Y”, but I don’t know how far they obstruct that.

Secondly, concerning testability, can one sample a random curve from this measure, for moderately large g (or even not so moderately large)? One thought would be to take to look at a high degree branched cover of P^1 and pick the ones defined over F_q, but this sounds uncomputable.

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