Étale cohomology and reduction of abelian varieties. (Q2773542)

Let \(X\) be a \(d\)-dimensional abelian variety over a field \(F\), and let \(\nu\) be a discrete valuation on \(F\). Let \(F^s\) denote a separable closure of \(F\), \(\overline\nu\) an extension of \(\nu\) to \(F^s\), and let \(I\) denote the inertia subgroup at \(\overline\nu\) of \(\text{Gal}(F^s/F)\). For a positive integer \(r\), let \(N(r)=\)\{prime powers \(\ell^m;0\leq m(\ell-1) \leq r\}\). The main result of this paper is as follows. Assume that \(0<k <2d\), \(k<r\), and \(n\notin N(r)\). When \(k\) is odd, \(X\) is semistable at \(\nu\) if and only if \((\sigma-1)^rH^k_{\text{ét}}(\overline X,\mathbb{Z}/n\mathbb{Z})=0\) for every \(\sigma\in I\). When \(k\) is even, \((\sigma-1)^r H^k_{\text{ét}} (\overline X,\mathbb{Z}/n\mathbb{Z})=0\) for every \(\sigma\in I\) if and only if either \(X\) is semistable at \(\nu\) or \(X\) has purely additive reduction at \(\nu\) but is semistable over a (ramified) quadratic extension of \(F\). Furthermore the authors show that one can shrink the exceptional set in the criteria when \(2\leq k\leq 2d-2\), and this exceptional set is minimal.