Congruences of Néron models for tori and the Artin conductor.

*(English)*Zbl 1098.14014Let \(K\) be a complete local field with a perfect residue field, and denote by \({\mathcal O}\) its ring of integers. It is generally known that every torus \(T\) over \(K\) has a canonical model \(\underline{T}^{\text{NR}}\) defined over \({\mathcal O}\) which is called its Néron model [cf. S. Bosch, W. Lütkebohmert and M. Raynaud, Néron models. Berlin etc.: Springer (1990; Zbl 0705.14001)]. This Néron model \(\underline{T}^{\text{NR}}\) of \(T\) is a smooth group scheme of finite type over the ring \({\mathcal O}\) such that \(\underline{T}^{\text{NR}}\otimes K\cong T\), and such that \(\underline{T}^{\text{NR}}\) admits a certain universal property with respect to morphisms of the form \(Z\otimes K\to T\), where \(Z\) is an arbitrary smooth scheme over \({\mathcal O}\). The group \(\Lambda:= X_*(T)\) of co-characters of \(T\) defined over \(K\) is a free abelian group of finite rank, equipped with a natural action of the Galois group \(\Gamma:= \text{Gal}(K^{\text{sep}}/K)\), and the torus \(T\) itself is known to be canonically determined by the lattice \(\Lambda\) and the action of \(\Gamma\) on it.

Now, via the theory of P. Deligne [in: Répresentations des Groupes Réductifs sur an Corps Local, Hermann. Paris, 119–157 (1984; Zbl 0578.12014)], for large ramification degree \(e\), the action of \(\Gamma\) on \(A\) factors through the Galois group \(\Gamma(\text{Tr}_eK)\) classifying extensions of the groundfield \(K\) that are at most \(e\)-ramified. Here \(\text{Tr}_eK:= ({\mathcal O}/p^e,p/p^e,\varepsilon)\) is the so-called “Deligne’s \(e\)th truncation” of the local ring \(({\mathcal O},p)\).

In this context, the paper under review investigates the naturally arising question of to what extent the pair \((\text{Tr}_eK,\Lambda)\) is related to the Néron model \(\underline{T}^{\text{NR}}\) of a given torus \(T\).

The authors’ main result (Theorem 9.2) confirms the expectation that the datum \((\text{Tr}_eK,\Lambda)\) determines the scheme \(\underline{T}^{\text{NR}}\otimes ({\mathcal O}/p^N)\) as \(e\gg N\). Actually, another version of this result is formulated without using Deligne’s theory (Theorem 8.5), and this statement is even valid for local fields that are not necessarily complete. As it is shown, in the sequel, the main result implies the equality of two invariants of tori over local fields as follows. On the one hand, one has the usual Artin conductor \(a(T)\) of the torus \(T\). On the other hand, B. Gross has constructed another important invariant \(c(T)\) using finite separable extensions \(L\) of the groundfield \(K\) [cf. B. H. Gross, Invent. Math. 130, No. 2, 287–313 (1997; Zbl 0904.11014); B. H. Gross and W. T. Gan, Trans. Am. Math. Soc. 351, No. 4, 1691–1704 (1999; Zbl 0991.20033)]. Combining their above-mentioned main result on the Néron model \(\underline{T}^{\text{NR}}\) of the torus \(T\) with both Gross’s theory and the Deligne-Kazhdan-Krasner approach to representing local fields in characteristic \(p> 0\) as limits of local fields in characteristic \(0\), the authors are able to establish the equality \(c(T)={1\over 2}a(T)\) for tori over an arbitrary local field. This extends partial results obtained by C.-L.Chai, B. H. Gross and E. Kushnirsky [cf.: E. Kushnirsky, Some arithmetical applications of integration on reductive groups, Ph. D. Thesis. University of Michigan, Ann Arbor (1999)] to the most general case. In an appendix to the present paper, the authors’ collaborator E. de Shalit adds an alternative proof of this equality of invariants in characteristic \(p> 0\). His proof is based on a finer analysis of the Lie algebra of the Néron model \(\underline{T}^{\text{NR}}\) on the one hand, and on methods of Galois cohomology once initiated by S. Sen [Ann. Math. (2) 90, 33–46 (1969; Zbl 0199.36301)], on the other hand.

Now, via the theory of P. Deligne [in: Répresentations des Groupes Réductifs sur an Corps Local, Hermann. Paris, 119–157 (1984; Zbl 0578.12014)], for large ramification degree \(e\), the action of \(\Gamma\) on \(A\) factors through the Galois group \(\Gamma(\text{Tr}_eK)\) classifying extensions of the groundfield \(K\) that are at most \(e\)-ramified. Here \(\text{Tr}_eK:= ({\mathcal O}/p^e,p/p^e,\varepsilon)\) is the so-called “Deligne’s \(e\)th truncation” of the local ring \(({\mathcal O},p)\).

In this context, the paper under review investigates the naturally arising question of to what extent the pair \((\text{Tr}_eK,\Lambda)\) is related to the Néron model \(\underline{T}^{\text{NR}}\) of a given torus \(T\).

The authors’ main result (Theorem 9.2) confirms the expectation that the datum \((\text{Tr}_eK,\Lambda)\) determines the scheme \(\underline{T}^{\text{NR}}\otimes ({\mathcal O}/p^N)\) as \(e\gg N\). Actually, another version of this result is formulated without using Deligne’s theory (Theorem 8.5), and this statement is even valid for local fields that are not necessarily complete. As it is shown, in the sequel, the main result implies the equality of two invariants of tori over local fields as follows. On the one hand, one has the usual Artin conductor \(a(T)\) of the torus \(T\). On the other hand, B. Gross has constructed another important invariant \(c(T)\) using finite separable extensions \(L\) of the groundfield \(K\) [cf. B. H. Gross, Invent. Math. 130, No. 2, 287–313 (1997; Zbl 0904.11014); B. H. Gross and W. T. Gan, Trans. Am. Math. Soc. 351, No. 4, 1691–1704 (1999; Zbl 0991.20033)]. Combining their above-mentioned main result on the Néron model \(\underline{T}^{\text{NR}}\) of the torus \(T\) with both Gross’s theory and the Deligne-Kazhdan-Krasner approach to representing local fields in characteristic \(p> 0\) as limits of local fields in characteristic \(0\), the authors are able to establish the equality \(c(T)={1\over 2}a(T)\) for tori over an arbitrary local field. This extends partial results obtained by C.-L.Chai, B. H. Gross and E. Kushnirsky [cf.: E. Kushnirsky, Some arithmetical applications of integration on reductive groups, Ph. D. Thesis. University of Michigan, Ann Arbor (1999)] to the most general case. In an appendix to the present paper, the authors’ collaborator E. de Shalit adds an alternative proof of this equality of invariants in characteristic \(p> 0\). His proof is based on a finer analysis of the Lie algebra of the Néron model \(\underline{T}^{\text{NR}}\) on the one hand, and on methods of Galois cohomology once initiated by S. Sen [Ann. Math. (2) 90, 33–46 (1969; Zbl 0199.36301)], on the other hand.

Reviewer: Werner Kleinert (Berlin)