Let \(D\) be a strictly henselian discrete valuation ring of residue characteristic \(p\geq 0\) and field of fractions \(K\). Let \(A\) be the Néron model of an abelian variety of dimension \(d\). Let \(\Phi\) denote the group of components of \(A\). While the \(p\)-part of this abelian group is still rather mysterious, its prime-to-\(p\) part \(\Phi_{(p)}\) is now well-understood thanks to the complete classification of the possible groups \(\Phi_{(p)}\) given in this article.
Let \(u,t\), and \(a\), denote respectively the unipotent, toric and abelian rank of \(A\). Let \(L/K\) denote the smallest Galois extension such that the generic fiber of \(A\) has semistable reduction, and let \(t_L\) and \(a_L\) denote the corresponding ranks. Recall that in the case of elliptic curves, one observes two distinct behaviours for the group \(\Phi\): If \(u=1\), then the order of \(\Phi\) is bounded by a constant depending only on \(u\) (namely, \(2^{2u})\), and if \(t=1\), then the group \(\Phi\) can be generated by a single element. The reviewer [J. Reine Angew. Math. 445, 109-160 (1993; Zbl 0781.14029)] generalized these facts by establishing, for arbitrary abelian varieties, the existence of a functorial filtration \((0)\subseteq \Phi^3_{(p)} \subseteq \Phi^2_{(p)} \subseteq \Phi^1_{(p)} \subseteq \Phi_{(p)}\), such that \(\Phi^3_{(p)}\) is generated by \(t\) elements, and \(|\Phi_{(p)}/ \Phi^2_{(p)} |\) and \(|\Phi^2_{(p)}/ \Phi^3_{(p)} |\) are bounded by explicit constants depending on \(t_L-t\) and \(a_L-a\).
In this article, the author improves these bounds as follows. If \(G\) is any abelian group and \(\ell\) is a prime, write the \(\ell\)-part \(G_\ell\) of \(G\) as \(G_\ell \cong\prod_{i\geq 1} \mathbb{Z}/ \ell^{m_{\ell,i}} \mathbb{Z}\), with \(m_{\ell,1}\geq m_{\ell,2} \geq\dots\). Let \(\delta_\ell(G): =\sum_{i\geq 1} (\ell^{m_{\ell,i}}-1)\), and let \(\delta(G) =\sum_\ell \delta_\ell(G)\). The author shows that \(\delta(\Phi^2_{(p)}/ \Phi^3_{(p)}) \leq t_L-t\), and that \(\delta(\Phi_{(p)}/ \Phi^2_{(p)}) \leq t_L-t+ 2(a_L-a)\). He then combines these two bounds into a single bound and proves the following theorem:
There exists an abelian variety over \(K\) of dimension \(d\) and ranks \(u,t\), and \(a\) which has \(\Phi_{(p)}\) isomorphic to a given group \(G=\prod_{\ell \neq p} G_\ell\) if and only if \[ 2u\geq \sum_{\ell\neq p} \sum_{i\geq t+1} (\ell^{\lfloor m_{\ell,i}/2 \rfloor}+ \ell^{\lceil m_{\ell,i}/2 \rceil}-2), \] where for any real number \(x\), \(\lfloor x\rfloor\) and \(\lceil x\rceil\) denote the largest integer \(\leq x\) (resp. the smallest integer \(\geq x)\).