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We study N\'eron models in arithmetic families over regular integral bases of finite type over \(S=\Spec\mathbb Z\), with \(\Spec\mathbb Z\) as the basic motivating case. The purpose is to isolate a common tame prime-to-residue-characteristic locus on which finite isogeny, component-group, inertia, Hecke, and height-theoretic data can be compared without confusing standard N\'eron-model input with the paper's organizational contribution. The method is to work after explicit shrinking to dense opens, interpret higher-dimensional bases through codimension-one DVRs, and combine N\'eron-model functoriality, finite-\`etale prime-to-\(p\) kernels (see \cite[Ch.~7, §7.3, Lem.~2, p.~179]{BLR}), Raynaud semistable reduction, and rational \(\ell\)-adic Tate-module comparison. The principal results show that, for a fixed finite class of degree-\(m\) prime-to-\(p\) isogenies, the induced maps on identity components have finite \`etale kernels killed by \(m\), while the maps on component groups have kernel and cokernel annihilated by a bounded power of \(m\). Consequently, the corresponding Tamagawa-index and local geometric defect terms are uniformly controlled on the chosen tame locus. The inertia-invariant subspaces of \(H^1_{\et}\) are preserved under these isogenies, and the Artin conductor exponent itself is unchanged because rational \(\ell\)-adic Tate modules of isogenous abelian varieties are isomorphic. For modular curves, and conditionally for Hodge-type Shimura varieties under stated integral-model extension hypotheses (\cite[Theorem~7.2.1, pp.~31--33]{PappasZachos2022}), the same mechanism yields Hecke-orbit uniformity away from the residue characteristic and ramified level primes. The height discussion records compatibility of standard semistable local height contributions involving \(\omega_{A/S}\), but does not assert a new global height gap. The practical implication is a precise, reusable tame framework for separating component-group variation from conductor invariance in arithmetic families.
Kundnani R. T., Kant Sh., Alam K., Marimuthu V. 2026. Prime-to-\(p\) Isogenies, Component Groups, and Cohomological Inertia for Néron Models. PREPRINTS.RU. https://doi.org/10.24108/preprints-3113861
