The Neron-Ogg-Shafarevich criterion in residue characteristic prime to l

Given: AC and DC, a discrete valuation ring $R$ with fraction field $K$, residue field $k$ and strict henselization $R^{\mathrm{s}\mathrm{h}}$, an abelian variety $A\mathrm{/}K$ of dimension $g$, its finite-type Neron model $N\mathrm{/}R$, and a prime $\mathrm{\ell }\mathrm{\ne }\mathrm{char}k$.

An abelian scheme over $R$ with generic fibre $A$ is a Neron model of $A$, and Neron models of smooth separated finite-type $K$-schemes are unique up to a unique $R$-isomorphism inducing the identity on generic fibres (An abelian scheme is the Neron model of its generic fibre, Uniqueness, weak Neron property, etale base change and local nature of Neron models).

For an abelian scheme $B\mathrm{/}R$ of relative dimension $g$ and $\mathrm{\ell }\mathrm{\ne }\mathrm{char}k$, each $B[{\mathrm{\ell }}^{\nu }]$ is finite etale over $R$ of rank ${\mathrm{\ell }}^{2g\nu }$, and over $R^{\mathrm{s}\mathrm{h}}$ specialization identifies its geometric generic points with its special separable points; the inertia group acts trivially on $B[{\mathrm{\ell }}^{\nu }]$ and on the Tate module (Abelian scheme torsion specialization is unramified, Prime to characteristic multiplication is etale).

For a field $F$ and $\mathrm{\ell }\mathrm{\ne }\mathrm{char}F$, $A[{\mathrm{\ell }}^{\nu }](F^{\mathrm{s}\mathrm{e}\mathrm{p}})\cong (\mathbb{Z}\mathrm{/}{\mathrm{\ell }}^{\nu }{)}^{2g}$ and $T_{\mathrm{\ell }}(A)$ is free of rank $2g$; an automorphism of $F^{\mathrm{s}\mathrm{e}\mathrm{p}}$ over $F$ acts trivially on $T_{\mathrm{\ell }}(A)$ if and only if it acts trivially on every $A[{\mathrm{\ell }}^{\nu }](F^{\mathrm{s}\mathrm{e}\mathrm{p}})$ (Field prime to characteristic torsion and Tate module, Prime-to-residue-characteristic Tate modules and inertia).

Over a strictly henselian local ring with separably closed residue field, a separated etale finite-type scheme $H$ satisfies $H(R)\cong H(k)$ by reduction. A Neron model has the extension property for etale local points; for $R^{\mathrm{s}\mathrm{h}}$ this follows by finite-stage approximation, and separatedness makes $N(R^{\mathrm{s}\mathrm{h}})\to A(K^{\mathrm{s}\mathrm{h}})$ bijective (Strict henselian etale sections, Uniqueness, weak Neron property, etale base change and local nature of Neron models, Neron models, the Neron mapping property and weak Neron models).

If $G$ is a smooth commutative finite-type group scheme of dimension $g$ over a field of characteristic $\mathrm{\ne }\mathrm{\ell }$ with $\mathrm{\mid }G[{\mathrm{\ell }}^{\nu }](\bar{k})\mathrm{\mid }={\mathrm{\ell }}^{2g\nu }$ for every $\nu \ge 1$, then $G^0$ is an abelian variety; a smooth separated finite-type quasi-projective $R$-scheme with abelian generic fibre and proper geometrically connected special fibre is proper over $R$ and an abelian scheme on its identity component; a smooth group scheme over a discrete valuation ring has an open identity component with connected generic and special fibres (Special fibre torsion growth detects properness, Connected smooth quasiprojective model with proper special fibre is proper, Divisor ampleness and quasi-projectivity of group models, The identity model of a smooth group with abelian generic fibre).

The Neron group scheme $N$ is commutative: its generic group law is commutative, and the two multiplication morphisms $N{\times }_{R}N\to N$ agree on the schematically dense generic fibre because $N$ is separated (Agreement on a schematically dense open). For a smooth commutative $R$-group scheme, if $n$ is a unit then $[n]:N\to N$ is etale: its differential at the identity is multiplication by $n$ on the locally free Lie module, and translations identify the differential at every point; apply the equal-relative-dimension criterion (Group schemes over a base scheme, Differentials of a smooth morphism, Dilatations and defect computation). Its kernel is therefore a separated etale finite-type $R$-scheme, though it need not be finite.