Suppose that \(K\) is a discretely valued extension of \(\mathbb{Q}_p\) and \(A_K\) is an abelian variety over \(K\) with semistable reduction. The Néron component group \(\Phi_{A_K}\) is the group of connected components of the special fibre of the Néron model. It is a finite abelian group and its \(l\)-primary part is computed in SGA7. The method fails for the \(p\)-primary part, because it describes an isomorphism between the unramified \(\mathbb{Z}_l{\operatorname{Gal}}(\overline K/K)\)-modules \(\Phi_{A_K}[l^\infty]\) and a piece of Galois cohomology. The latter is “too small” at the \(p\)-torsion part, but M. Kim and S. H. Marshall [Math. Res. Lett. 7, No. 5–6, 605–614 (2000; Zbl 0979.11034)] fixed this by instead looking at the functor \({\operatorname{Crys}}_1\) that takes a \(p\)-adic Galois module to its maximal \(1\)-crystalline submodule, and applying the derived functor of that to the Tate module \(T_pA_K\). Their method works for \(p\) odd, and as long as \(K\) is an unramified extension, and the effect of this paper is to remove those restrictions.
To do this, the author finds, as Kim and Marshall did, a finite flat \({\mathcal O}_K\)-group \(Q_{p^m}\) whose generic fibre is \({\operatorname{Crys}}_1(A_K[p^m])\), but by a more direct and constructive approach using the Néron model.
Crucial ingredients in the rather technical proof are the results of M. Kim and S. H. Marshall [Math. Res. Lett. 7, No. 5–6, 605–614 (2000; Zbl 0979.11034)], a full faithfulness result due to Y. Ozeki [Nagoya Math. J. 229, 169–214 (2018; Zbl 1441.11138)], and more generally the background as expounded by S. H. Marshall in her PhD thesis [Crystalline representations and Néron models. Tucson: University of Arizona (PhD Thesis) (2001)].