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# Group of connected components of the global N\'eron-RaynaudNéron-Raynaud model of a torus

Hi.

Let $K = \mathbb{F}_q(C)$ be a global function field of an irreducible projective and smooth curve $C$ defined over a finite field of constants $\mathbb{F}_q$. Let $T$ be a $K$-torus. We choose one closed point $\infty$ to be the point at infinity, and consider the ring of ${\infty}$-integers of $K$, namely: $$A_\infty = \{ a \in K : v_\frak{p}(a) \geq 0 \ \ \forall \frak{p} \neq \infty \}.$$ Then $T$ admits a global N\'eron-Raynaud Néron-Raynaud model $\mathcal{T}$ defined over $\text{Spec}~A_\infty$, which is of finite type, obtained by glueing all local N\'eron-Raynaud Néron-Raynaud models $\mathcal{T}_\frak{p}$ which are of finite type, defined each one over the corresponding local valuation ring $\mathcal{O}_p$. The glueing is along the generic fiber $T$. Denote by $\mathcal{T}_\frak{p}$ a connected reduction modulo $\frak{p}$. Let $\mathcal{T}^0$ denote the subscheme of $\mathcal{T}$ whose geometric fibers are $\mathcal{T}_\frak{p}^0$.

My question is can I express the finite index $[\mathcal{T}(A_\infty):\mathcal{T}^0(A_\infty)]$ as the product of $[\mathcal{T}_\frak{p}(\mathcal{O}_\frak{p}):\mathcal{T}_\frak{p}^0(\mathcal{O}_p)]$ (which may differ from 1 only if $\frak{p}$ is ramified on the minimal splitting field of $T$) running over all $\frak{p} \neq \infty$ ?

Thank you, Rony.

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Hi.

Let $K = \mathbb{F}_q(C)$ be a global function field of an irreducible projective and smooth curve $C$ defined over a finite field of constants $\mathbb{F}_q$. Let $T$ be a $K$-torus. We choose one closed point $\infty$ to be the point at infinity, and consider the ring of ${\infty}$-integers of $K$, namely: $$A_\infty = \{ a \in K : v_\frak{p}(a) \geq 0 \ \ \forall \frak{p} \neq \infty \}.$$ Then $T$ admits a global N\'eron-Raynaud model $\mathcal{T}$ defined over $\text{Spec}~A_\infty$, which is of finite type, obtained by glueing all local N\'eron-Raynaud models $\mathcal{T}_\frak{p}$ which are of finite type, defined each one over the corresponding local valuation ring $\mathcal{O}_p$. The glueing is along the generic fiber $T$. Denote by $\mathcal{T}_\frak{p}$ a connected reduction modulo $\frak{p}$. Let $\mathcal{T}^0$ denote the subscheme of $\mathcal{T}$ whose geometric fibers are $\mathcal{T}_\frak{p}^0$.

My question is can I express the finite index $[\mathcal{T}(A_\infty):\mathcal{T}^0(A_\infty)]$ in terms of as the product of $[\mathcal{T}_\frak{p}(\mathcal{O}_\frak{p}):\mathcal{T}_\frak{p}^0(\mathcal{O}_p)]$ (which may differ from 1 only if $\frak{p}$ is ramified on the minimal splitting field of $T$) running over all $\frak{p} \neq \infty$ ?

Thank you, Rony.

Hi.

Let $K = \mathbb{F}_q(C)$ be a global function field of an irreducible projective and smooth curve $C$ defined over a finite field of constants $\mathbb{F}_q$. Let $T$ be a $K$-torus. We choose one closed point $\infty$ to be the point at infinity, and consider the ring of ${\infty}$-integers of $K$, namely: $$A_\infty = \{ a \in K : v_\frak{p}(a) \geq 0 \ \ \forall \frak{p} \neq \infty \}.$$ Then $T$ admits a global N\'eron-Raynaud model $\mathcal{T}$ defined over $\text{Spec}~A_\infty$, which is of finite type, obtained by glueing all local N\'eron-Raynaud models $\mathcal{T}_\frak{p}$ which are of finite type, defined each one over the corresponding local valuation ring $\mathcal{O}_p$. The glueing is along the generic fiber $T$. Denote by $\mathcal{T}_\frak{p}$ a connected reduction modulo $\frak{p}$. Let $\mathcal{T}^0$ denote the subscheme of $\mathcal{T}$ whose geometric fibers are $\mathcal{T}_\frak{p}^0$.

My question is can I express the finite index $[\mathcal{T}(A_\infty):\mathcal{T}^0(A_\infty)]$ in terms of the product of $[\mathcal{T}_\frak{p}(\mathcal{O}_\frak{p}):\mathcal{T}_\frak{p}^0(\mathcal{O}_p)]$ (which may differ from 1 only if $\frak{p}$ is ramified on the minimal splitting field of $T$) running over all $\frak{p} \neq \infty$ ?

Thank you, Rony.

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