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On Tate–Shafarevich groups of one-dimensional families of commutative group schemes over number fields

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Abstract

Given a smooth geometrically connected curve C over a field k and a smooth commutative group scheme G of finite type over the function field K of C, we study the Tate–Shafarevich groups given by elements of \(H^1(K,G)\) locally trivial at completions of K associated with closed points of C. When G comes from a k-group scheme and k is a number field (or k is a finitely generated field and C has a k-point), we prove finiteness of generalizing a result of Saïdi and Tamagawa for abelian varieties. We also give examples of nontrivial in the case when G is a torus and prove other related statements.

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References

  1. Colliot-Thélène, J.-L., Sansuc, J.-J.: La R-équivalence sur les tores. Ann. Sci. École Norm. Sup. 10, 175–229 (1977)

    Article  MathSciNet  Google Scholar 

  2. Colliot-Thélène, J.-L., Skorobogatov, A.N.: The Brauer–Grothendieck group. Springer, Cham (2021)

    Book  Google Scholar 

  3. Harari, D.: Méthode des fibrations et obstruction de Manin. Duke Math. J. 75, 221–260 (1994)

    Article  MathSciNet  Google Scholar 

  4. Harari, D.: Galois cohomology and class field theory, Springer-Verlag–EDP Sciences (2020)

  5. Harari, D., Szamuely, T.: Arithmetic duality theorems for 1-motives. J. Reine Angew. Math. 578, 93–128 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Lichtenbaum, S.: Duality theorems for curves over \(p\)-adic fields. Invent. Math. 7, 120–136 (1969)

    Article  MathSciNet  Google Scholar 

  7. Milne, J.S.: Arithmetic Duality Theorems, 2nd edn. BookSurge, LLC, Charleston, SC (2006)

  8. Nisnevich, Y.A.: Espaces homogènes principaux rationnellement triviaux et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind. C. R. Acad. Sci. Paris 299, 5–8 (1984)

    MathSciNet  MATH  Google Scholar 

  9. Rapinchuk, A., Rapinchuk, I.: Some finiteness results for algebraic groups and unramified cohomology over higher-dimensional fields. J. Number Theory 233, 228–260 (2022)

    Article  MathSciNet  Google Scholar 

  10. Saïdi, M., Tamagawa, A.: On the arithmetic of abelian varieties. J. Reine Angew. Math. 762, 1–33 (2020)

    Article  MathSciNet  Google Scholar 

  11. Sansuc, J.-J.: Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres. J. Reine Angew. Math. 327, 12–80 (1981)

    MathSciNet  MATH  Google Scholar 

  12. Scheiderer, C., van Hamel, J.: Cohomology of tori over \(p\)-adic curves. Math. Ann. 326, 155–183 (2003)

    Article  MathSciNet  Google Scholar 

  13. Voskresenskiĭ, V.E.: Birational properties of linear algebraic groups. Izv. Akad. Nauk SSSR Ser. Mat. 34, 3–19 (1970)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405, Orsay, France

    David Harari

  2. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Tamás Szamuely

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  1. David Harari
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  2. Tamás Szamuely
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Correspondence to David Harari.

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Harari, D., Szamuely, T. On Tate–Shafarevich groups of one-dimensional families of commutative group schemes over number fields. Math. Z. 302, 935–948 (2022). https://doi.org/10.1007/s00209-022-03080-x

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  • DOI: https://doi.org/10.1007/s00209-022-03080-x

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