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${\overline {\mathcal {M}}}_{g,n}$
We study collections of subrings of 
$H^*({\overline {\mathcal {M}}}_{g,n})$ that are closed under the tautological operations that map cohomology classes on moduli spaces of smaller dimension to those on moduli spaces of larger dimension and contain the tautological subrings. Such extensions of tautological rings are well-suited for inductive arguments and flexible enough for a wide range of applications. In particular, we confirm predictions of Chenevier and Lannes for the 
$\ell $-adic Galois representations and Hodge structures that appear in 
$H^k({\overline {\mathcal {M}}}_{g,n})$ for 
$k = 13$, 
$14$ and 
$15$. We also show that 
$H^4({\overline {\mathcal {M}}}_{g,n})$ is generated by tautological classes for all g and n, confirming a prediction of Arbarello and Cornalba from the 1990s. In order to establish the final base cases needed for the inductive proofs of our main results, we use Mukai’s construction of canonically embedded pentagonal curves of genus 7 as linear sections of an orthogonal Grassmannian and a decomposition of the diagonal to show that the pure weight cohomology of 
${\mathcal {M}}_{7,n}$ is generated by algebraic cycle classes, for 
$n \leq 3$.
$\overline {\mathcal {M}}_{g,n}$
We prove that the rational cohomology group 
$H^{11}(\overline {\mathcal {M}}_{g,n})$ vanishes unless 
$g = 1$ and 
$n \geq 11$. We show furthermore that 
$H^k(\overline {\mathcal {M}}_{g,n})$ is pure Hodge–Tate for all even 
$k \leq 12$ and deduce that 
$\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$ is surprisingly well approximated by a polynomial in q. In addition, we use 
$H^{11}(\overline {\mathcal {M}}_{1,11})$ and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and nontautological algebraic cycle classes in Chow cohomology.
Let 
$C$ be a curve over a complete valued field having an infinite residue field and whose skeleton is a chain of loops with generic edge lengths. We prove that any divisor on the chain of loops that is rational over the value group lifts to a divisor of the same rank on $C$ , confirming a conjecture of Cools, Draisma, Robeva, and the third author.
