Survey of progress verifying Hodge and Tate conjectures for moduli spaces of curves via inductive boundary stratification.
Moduli spaces of curves with polynomial point counts
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abstract
We prove that the number of curves of a fixed genus g over finite fields is a polynomial function of the size of the field if and only if g is at most 8. Furthermore, we determine for each positive genus g the smallest n such that the moduli space of curves of genus g with n marked points does not have polynomial point count. A key ingredient in the proofs, which is also a new result of independent interest, is the computation of the thirteenth cohomology group of the moduli spaces of stable curves of genus g with n marked points, for all g and n.
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math.AG 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Expository survey of when Chow rings and cohomology rings of moduli spaces of curves are tautological and when their point counts over finite fields are polynomials in q.
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On the Hodge and Tate conjectures for moduli spaces of curves
Survey of progress verifying Hodge and Tate conjectures for moduli spaces of curves via inductive boundary stratification.
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Chow rings, cohomology rings, and point counts of moduli spaces of curves
Expository survey of when Chow rings and cohomology rings of moduli spaces of curves are tautological and when their point counts over finite fields are polynomials in q.