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Colliot-Thélène's Hasse principle conjecture for zero-cycles

Let KK be a global field, and let V/KV/K be a smooth projective variety such that its base change VK‾=V⊗⁡K‾V_{\overline K}=V\operatorname*{\otimes}\overline K is rational and has Picard group Pic⁡(VK‾)=Z\operatorname{Pic}(V_{\overline K})=\mathbf Z. For each place ν\nu of KK, let VKν=V⊗⁡KKνV_{K_{\nu}}=V\operatorname*{\otimes}_K K_{\nu}. A 00-cycle of degree 11 means a zero-cycle whose degree over the relevant field is 11. Colliot-Thélène's conjecture. If VKνV_{K_{\nu}} has a 00-cycle of degree 11 for every place ν\nu of KK, then VV has a 00-cycle of degree 11. This is stated as a special case of the broader conjecture that the Brauer–Manin obstruction is the only obstruction to zero-cycles of degree 11 on smooth proper varieties over global fields; the supplied source gives no resolution status.

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  1. Colliot-Thélène's Hasse principle conjecture for zero-cycles

    Let KK be a number field, let AA be its ring of adeles, and let XX be a proper variety over KK. Write X(A)Br⁡(X)X(A)^{\operatorname{Br}(X)} for the subset of adelic points annihilated by the Brauer–Manin pairing. A zero-cycle on XX has degree 11 if the sum of the degrees of its closed points, counted with multiplicity, is 11. Colliot-Thélène's Hasse principle conjecture. If

    X(A)Br⁡(X)≠∅,X(A)^{\operatorname{Br}(X)} \ne \emptyset,

    then there is a zero-cycle of degree 11 over KK on XX. This is the Hasse principle for zero-cycles. Saito proved the conjecture for curves assuming finiteness of the Tate–Shafarevich group; the general case remains open, and it is not known whether Skorobogatov's counterexample to the corresponding rational-point question has a zero-cycle of degree 11.

    source: Max Lieblich, “Arithmetic aspects of moduli of sheaves on curves”, arXiv:0902.0482 (2009).

Sources & referencesView supporting material

Primary source

Max Lieblich, “Period and index in the Brauer group of an arithmetic surface (with an appendix by Daniel Krashen)”, arXiv:math/0702240 (2009).

arithmetic-geometrybrauer-manin-obstructionzero-cycles

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