[Paper Review] On Neron-Raynaud class groups of tori and the Capitulation Problem

Cristian D. González-Avilés|ArXiv.org|Mar 18, 2009

Geometric and Algebraic Topology15 references3 citations

TL;DR

This paper generalizes the classical capitulation problem for ideal class groups to Néron-Raynaud class groups of tori over global fields using cohomological methods. It establishes long exact sequences linking these class groups to Galois cohomology and ideal class groups, particularly resolving the class group of a torus split by a metacyclic extension in terms of classical ideal class groups.

ABSTRACT

We discuss the Capitulation Problem for Neron-Raynaud class groups of tori over global fields F and obtain generalizations of the main results of [10]. We also show that short exact sequences of F-tori induce long exact sequences involving the corresponding Neron-Raynaud class groups. For example, we show that the Neron-Raynaud class group of any F-torus which is split by a metacyclic extension of F can be "resolved" in terms of classical ideal class groups of global fields.

Motivation & Objective

• To extend the classical S-capitulation problem for ideal class groups to Néron-Raynaud class groups of tori over global fields.
• To establish long exact sequences involving Néron-Raynaud class groups of F-tori via Galois cohomology.
• To resolve the Néron-Raynaud class group of a torus split by a metacyclic extension in terms of classical ideal class groups.
• To provide a cohomological interpretation of class groups using Nisnevich cohomology, simplifying adelic definitions.
• To compute the Ono invariant of norm tori in terms of class groups and norm maps under coflasque assumptions.

Proposed method

• Utilizes Nisnevich cohomology to provide a non-adelic, cohomologically interpretable description of Néron-Raynaud class groups.
• Applies Galois cohomology to study the S-capitulation map $ j_{T,K/F,S} $, analyzing its kernel and cokernel via exact sequences.
• Introduces the group $ H^1(G, \widetilde{\mathcal{T}}^\circ(\widetilde{U}))' $ as a key object in the kernel description.
• Uses localization maps $ \lambda_S $ to relate global cohomology to local cohomology at primes outside $ S $.
• Applies the norm torus construction $ T' = R^{(1)}_{K/F}(T_K) $ to reduce the problem to coflasque resolutions.
• Employs the exact sequence involving $ \operatorname{Ker} N_{\mathcal{O}} $, $ C_{T',F,S} $, and $ C_{T',F,S}^* $ to derive formulas for Ono invariants.

Experimental results

Research questions

• RQ1 How can the classical capitulation problem for ideal class groups be generalized to Néron-Raynaud class groups of tori?
• RQ2 What is the structure of the kernel and cokernel of the S-capitulation map $ j_{T,K/F,S} $ for tori over global fields?
• RQ3 Under what conditions can the Néron-Raynaud class group of a torus be resolved in terms of classical ideal class groups?
• RQ4 How does the Ono invariant of a norm torus relate to Galois cohomology and norm maps on Néron-Raynaud models?
• RQ5 What is the role of the Nisnevich cohomological interpretation in simplifying the study of class groups of tori?

Key findings

• A canonical exact sequence is established: $ 0 \to \operatorname{Ker} j_{T,K/F,S} \to H^1(G, \widetilde{\mathcal{T}}^\circ(\widetilde{U}))' \xrightarrow{\lambda_S} \bigoplus_{v \notin S} H^1(G_{w_v}, \widetilde{\mathcal{T}}^\circ(\mathcal{O}_{w_v}))' \to \operatorname{Coker} j_{T,K/F,S}' \to 0 $, generalizing [10, Theorem 2.4].
• When $ T $ splits over $ K $, a longer exact sequence involving $ H^2(G, \widetilde{\mathcal{T}}^\circ(\widetilde{U})) $, $ B_S(G,T) $, and $ H^3(G, \widetilde{\mathcal{T}}^\circ(\widetilde{U})) $ describes the cokernel of $ j_{T,K/F,S} $, generalizing [10, Theorem 3.3].
• For a coflasque torus $ T $ with $ T_K $ quasi-trivial, the Ono invariant is given by $ E_{c,S}(T') = \frac{[\operatorname{Sh}_{N,S}(T)]}{[\mathcal{T}^\circ(U) : N_{K/F}\widetilde{\mathcal{T}}^\circ(\widetilde{U})] \cdot [C_{T,F,S} : N_{T,K,S_{K}} C_{T,K,S_K}]} $, providing a cohomological formula.
• The kernel of the norm map $ N_{\mathcal{O}}: \widetilde{\mathcal{T}}^\circ(\widetilde{U}) \to \mathcal{T}^\circ(U) $ is shown to be isomorphic to $ \operatorname{Ker} \vartheta_{T',S}^* $, linking it to class groups.
• An exact sequence is derived: $ 0 \to W_{T,F,S}/N_{K/F}\widetilde{\mathcal{T}}^\circ(\widetilde{U}) \to C_{T',F,S} \to C_{T,K,S_K} \xrightarrow{\nu} C_{T,F,S}^N \to 0 $, where $ W_{T,F,S} $ is a group defined via local cohomology.
• The group $ H^1(F_v^{\rm{nr}}, T')^{G_{k(v)}} $ is identified with the torsion subgroup of the component group $ \Phi_v(T')(k(v)) $, extending results from [11, §4].