Cohomological Detection of Good Reduction on Tame Moduli Stacks: Nearby Cycles, $N=0$, and a Stacky Néron--Ogg--Shafarevich Criterion

Le résumé fourni par la source

Assuming purity and tame monodromy of nearby cycles on algebraic stacks and the existence of smooth local slice models of abelian or K3 type, we establish a cohomological criterion for detecting good reduction in families parametrized by tame moduli stacks. Under these hypotheses, trivial monodromy on $\ell$-adic cohomology implies the integral extendability of points with finite stabilizers, generalizing the Néron–Ogg–Shafarevich criterion from abelian varieties to stacks with linearly reductive inertia. The theory yields density results for good-reduction loci, clarifies the role of weight–monodromy and nearby-cycle purity in extension problems, and provides explicit examples and counterexamples illustrating the necessity of the hypotheses. Applications include elliptic curves with level structure, abelian surfaces with complex multiplication, and tame quotient stacks $[\A^1/\mu_n]$ with $(n,p)=1$, linking cohomological purity to arithmetic integrality and moduli-theoretic extension properties.

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