The finiteness of the Tate–Shafarevich group over function fields for algebraic tori defined over the base field

Rapinchuk, Andrei; Rapinchuk, Igor

Comptes Rendus. Mathématique, Volume 362 (2024) no. G7, pp. 739-749

Let $K$ be a field and $V$ be a set of rank one valuations of $K$. The corresponding Tate–Shafarevich group of a $K$-torus $T$ is $Ш(T,V)=ker(H^1(K,T)\to {\prod }_{v\in V}{H}^1(K_v,T))$. We prove that if $K=k\left(X\right)$ is the function field of a smooth geometrically integral quasi-projective variety over a field $k$ of characteristic 0 and $V$ is the set of discrete valuations of $K$ associated with prime divisors on $X$, then for any torus $T$ defined over the base field $k$, the group $Ш(T,V)$ is finite in the following situations: (1) $k$ is finitely generated and $X\left(k\right)\ne \varnothing$; (2) $k$ is a number field.

Soit $K$ un corps et $V$ un ensemble de valuations de rang un de $K$. Le groupe de Tate–Shafarevich d’un $K$-tore T est $Ш(T,V)=ker(H^1(K,T)\to {\prod }_{v\in V}{H}^1(K_v,T))$. Nous montrons que si $K=k\left(X\right)$ est le corps de fonctions d’une variété lisse géométriquement intègre quasi-projective définie sur un corps $k$ de caractéristique 0 et que $V$ est l’ensemble des valuations discrètes de $K$ associées aux diviseurs irréductibles de $X$, alors pour tout tore $T$ défini sur le corps de base $k$, le groupe $Ш(T,V)$ est fini dans les situations suivantes  : (1) $k$ est de type fini sur le corps premier et $X\left(k\right)\ne \varnothing$ ; (2) $k$ est un corps de nombres.

[1] Conrad, B. Deligne’s notes on Nagata compactifications, J. Ramanujan Math. Soc., Volume 22 (2007) no. 3, pp. 205-257 | Zbl | MR

[2] Colliot-Thélène, J.-L.; Sansuc, J.-J. Principal homogeneous spaces under flasque tori; applications, J. Algebra, Volume 106 (1987), pp. 148-205 | DOI | Zbl | MR

[3] Fedorov, R.; Panin, I. A proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing infinite fields, Publ. Math., Inst. Hautes Étud. Sci., Volume 122 (2015), pp. 169-193 | DOI | Zbl | Numdam | MR

[4] Guralnick, R.; Jaffe, D. B.; Raskind, W.; Wiegand, R. On the Picard group: Torsion and the kernel induced by a faithfully flat map, J. Algebra, Volume 183 (1996) no. 2, pp. 420-455 | DOI | Zbl | MR

[5] Ghosh, M.; Krishna, A. Bertini theorems revisited, J. Lond. Math. Soc., Volume 108 (2023) no. 3, pp. 1163-1192 | DOI | Zbl | MR

[6] Guo, N. The Grothendieck–Serre conjecture over semilocal Dedekind rings, Transform. Groups, Volume 27 (2022) no. 3, pp. 897-917 | DOI | Zbl | MR

[7] Görtz, U.; Wedhorn, T. Algebraic geometry I. Schemes. With examples and exercises, Springer Studium Mathematik – Master, Springer, 2020 | Zbl | DOI | MR

[8] Hartshorne, R. Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977 | Zbl | DOI | MR

[9] Harari, D.; Szamuely, T. On Tate–Shafarevich groups of one-dimensional families of commutative group schemes over number fields, Math. Z., Volume 302 (2022) no. 2, pp. 935-948 | DOI | Zbl | MR

[10] Kleiman, S. L.; Altman, A. B. Bertini theorems for hypersurface sections containing a subscheme, Commun. Algebra, Volume 7 (1979), pp. 775-790 | DOI | Zbl | MR

[11] Kahn, B. On the class group of an arithmetic scheme, Bull. Soc. Math. Fr., Volume 134 (2006) no. 3, pp. 395-415 | DOI | Zbl | MR

[12] Lang, S.; Weil, A. Number of points of varieties in finite fields, Am. J. Math., Volume 76 (1954), pp. 819-827 | DOI | Zbl | MR

[13] Lütkebohmert, W. On compactification of schemes, Manuscr. Math., Volume 80 (1993) no. 1, pp. 95-111 | DOI | Zbl | MR

[14] Milne, J. S. Class Field Theory (available on the author’s website at https://www.jmilne.org/math/CourseNotes/cft.html)

[15] Nisnevich, Y. A. Rationally trivial principal homogeneous spaces and arithmetic of reductive group schemes over Dedekind rings, C. R. Math. Acad. Sci. Paris, Volume 299 (1984), pp. 5-8 | Zbl

[16] Platonov, V.; Rapinchuk, A. Algebraic groups and number theory, Pure and Applied Mathematics, 139, Academic Press Inc., 1994 (transl. from the russian by Rachel Rowen) | Zbl

[17] Rapinchuk, I. A. Simple algebraic groups with good reduction, the genus problem, and unramified cohomology (in preparation)

[18] Rapinchuk, A. S.; Rapinchuk, I. A. Linear algebraic groups with good reduction, Res. Math. Sci., Volume 7 (2020) no. 3, 28 | DOI | Zbl | MR

[19] Rapinchuk, A. S.; Rapinchuk, I. A. Finiteness theorems for algebraic tori over function fields, C. R. Math. Acad. Sci. Paris, Volume 359 (2021) no. 8, pp. 939-944 | DOI | Zbl | Numdam | MR

[20] Rapinchuk, A. S.; Rapinchuk, I. A. Some finiteness results for algebraic groups and unramified cohomology over higher-dimensional fields, J. Number Theory, Volume 233 (2022), pp. 228-260 | DOI | Zbl | MR

[21] Rapinchuk, A. S.; Rapinchuk, I. A. Properness of the global-to-local map for algebraic groups with toric connected component and other finiteness properties, Math. Res. Lett., Volume 30 (2023) no. 3, pp. 913-943 | DOI | Zbl | MR

[22] Serre, J.-P. Galois cohomology, Springer, 1997 | DOI | MR

[23] Voskresenskiĭ, V. E. Algebraic Groups and Their Birational Invariants, Translations of Mathematical Monographs, 179, American Mathematical Society, 1998 | Zbl | MR

[24] The Stacks Project Authors Stacks Project (https://stacks.math.columbia.edu)