Finitude du groupe de Tate-Shafarevich pour les groupes de type multiplicatif constants sur des corps des fonctions
Melvyn El Kamel-Meyrigne
TL;DR
This work establishes finiteness of Tate–Shafarevich groups $\Sha^1(K,G)$ for multiplicative-type groups $G$ over a function-field setup $K/k_0((x_1,...,x_n))$, via a careful comparison with cohomology on a desingularized model $\mathcal X$ of $\operatorname{Spec}(R)$. The approach reduces the global obstruction to local data on the special fiber, controlled by two key hypotheses about kernels in Galois cohomology and Picard groups, and uses a split-case analysis together with Picard and Mordell–Weil-type finiteness results. When $R$ is regular, the obstruction vanishes and $\Sha^1(K,G)=0$, highlighting a strong local-global principle in this regular setting. The paper also sketches extensions of these results to residue fields finitely generated over $\mathbb{Q}$ and to function fields over $\mathbb{Q}((x))$, indicating broad applicability of the method to other base fields. The results contribute to the understanding of local-global principles for torsors under tori and multiplicative-type groups over higher-dimensional function fields.
Abstract
Let $k_0$ be a number field, $K$ be a finite extension of $k_0(\!(x_1,...,x_n)\!)$ and let $R$ be the integral closure of $k_0[[x_1,...,x_n]]$ in $K$. Consider a group of multiplicative type $G$ defined over $K$. We study the Tate-Shafarevich group given by elements of $H^1(K,G)$ locally trivial at completions of $K$ with respect to the points of codimension $1$ of $Spec(R)$. We show the finiteness of the Tate-Shafarevich group when $G$ comes from a group of multiplicative type $G_{k_0}$ defined over $k_0$ provided that two technical conditions are satisfied. We then prove that the Tate-Shafarevich group is trivial when the ring of integers $R$ is regular. -- Soient $k_0$ un corps de nombres, $K$ une extension finie de $k_0(\!(x_1,...,x_n)\!)$ et soit $R$ la clôture intégrale de $k_0[[x_1,...,x_n]]$ dans $K$. Soit $G$ un groupe de type multiplicatif défini sur $k_0$. On étudie le groupe de Tate-Shafarevich donné par les éléments de $H^1(K,G)$ localement triviaux aux complétions de $K$ par rapport aux points de codimension $1$ de $Spec(R)$. On établit la finitude du groupe de Tate-Shafarevich lorsque $G$ provient d'un groupe de type multiplicatif $G_{k_0}$ défini sur $k_0$ sous réserve que deux hypothèses techniques soient satisfaites. On montre que le groupe de Tate-Shafarevich est trivial lorsque l'anneau des entiers $R$ est régulier.