Fixed $q$

The results of Katz and Sarnak [ MR 2000b:11070] concern curves of genus $g$ over ${\mathbb F}_q$ with both $g\to\infty$ and $q\to\infty$.

It would be interesting to have results in the case of fixed $q$. Does one still get random matrix statistics?

A particular case that it worth investigating is the set of all quadratics over ${\mathbb F}_q$.

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