Special fibre torsion growth detects properness

Over $\bar{k}$ write $0\to N\to G_{\bar{k}}^0\to B\to 0$ with $N$ smooth connected affine of dimension $a$ and $B$ an abelian variety of dimension $b$, so that $g=a+b$; this is the Barsotti-Chevalley decomposition of [F1]. The torsion of $G_{\bar{k}}^0[{\mathrm{\ell }}^{\nu }]$ maps to $B[{\mathrm{\ell }}^{\nu }]$ with fibres that are torsors under $N[{\mathrm{\ell }}^{\nu }]$, of cardinal at most $\mathrm{\mid }N[{\mathrm{\ell }}^{\nu }]\mathrm{\mid }\le {\mathrm{\ell }}^{a\nu }$ by [F1], while $\mathrm{\mid }B[{\mathrm{\ell }}^{\nu }]\mathrm{\mid }={\mathrm{\ell }}^{2b\nu }$ by [F2]. Hence $\mathrm{\mid }{G}^0[{\mathrm{\ell }}^{\nu }](\bar{k})\mathrm{\mid }\le {\mathrm{\ell }}^{(2b+a)\nu }={\mathrm{\ell }}^{(2g-a)\nu }$.

The group $G$ has finitely many connected components, say $C$ of them, and translation by a torsion point in a component embeds that component's torsion into $G^0[{\mathrm{\ell }}^{\nu }]$ (subtracting a torsion point identifies the component with $G^0$ and preserves torsion), so $\mathrm{\mid }G[{\mathrm{\ell }}^{\nu }](\bar{k})\mathrm{\mid }\le C\cdot \mathrm{\mid }{G}^0[{\mathrm{\ell }}^{\nu }](\bar{k})\mathrm{\mid }\le C{\mathrm{\ell }}^{(2g-a)\nu }$. Comparing with the hypothesis ${\mathrm{\ell }}^{2g\nu }$ gives ${\mathrm{\ell }}^{a\nu }\le C$ for all $\nu \ge 1$, which forces $a=0$.

Therefore $N$ is trivial and $G_{\bar{k}}^0=B$ is an abelian variety. By [F2], each $G[{\mathrm{\ell }}^{\nu }]$ is finite etale, so passage from $k^{\mathrm{s}\mathrm{e}\mathrm{p}}$ to $\bar{k}$ changes no torsion points; the count is therefore already available over $k^{\mathrm{s}\mathrm{e}\mathrm{p}}$. Geometric properness of $G^0$ descends from $\bar{k}$ to $k$ by [F2], so $G^0$ is proper over $k$ and, being smooth connected commutative and proper, is an abelian variety over $k$. This bound suffices for the arithmetic applications and avoids asserting a stronger exact exponent.