Moduli spaces of curves with polynomial point counts

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.

Figures (3)

• Figure 1: The table shows $\frac{1}{2} \chi^{\mathbb{S}}_{13}(\mathcal{M}_{g,n})$.
• Figure 2: Plot of the ratio $Z_g/Z_g^{asymp}$ for $g$ between 30 and 2400. Note in particular that $Z_g$ is not zero in this range. The two curves above correspond to whether $g$ is even or odd: in addition to the formula for $Z_g^{asymp}$ depending on the parity of $g$, the convergence rates are different depending on the parity of $g$. The Euler characteristics for $g\leq 30$ can be found in PayneWillwacher24.
• Figure 3: The plot shows $\chi_{11}(\mathcal{M}_{g,n})\frac{ (-1)^{g(g-1)/2} (2\pi)^g }{2(n+g-2)!}$ for $n=20$ (green) and $n=30$ (blue).

Theorems & Definitions (99)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.4
• Theorem 1.5
• Conjecture 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Remark 1.10
• Remark 1.11