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8
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Group of connected components of the global N\'eron-Raynaud Néron-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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7
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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.
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6
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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 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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5
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Hi.
Let $K = F_q(C)$ \mathbb{F}_q(C)$ be a global function field of an irreducible projective and smooth curve $C$
defined over a finite field of constants $F_q$. \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 $\mathcal{T}_p$
\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 $\mathcal{T}_p^0$ the open subscheme
of $\mathcal{T}_p$ having \mathcal{T}_\frak{p}$ a connected reduction modulo $p$.
\frak{p}$.
Let $\mathcal{T}^0$ denote the subscheme
of $\mathcal{T}$ whose geometric fibers are $\mathcal{T}_p^0$. \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 $[\mathcal{T}_p(\mathcal{O}_p):\mathcal{T}_p^0(\mathcal{O}_p)]$
[\mathcal{T}_\frak{p}(\mathcal{O}_\frak{p}):\mathcal{T}_\frak{p}^0(\mathcal{O}_p)]$
(which may differ from 1 only if $p$ \frak{p}$ is ramified on the minimal splitting field of $T$) ?
Thank you, Rony.
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4
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Hi.
Let $K = F_q(C)$ be a global function field of an irreducible projective and smooth curve $C$
defined over a finite field of constants $F_q$. 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 $\mathcal{T}_p$
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 $\mathcal{T}_p^0$ the open subscheme
of $\mathcal{T}_p$ having a connected reduction modulo $p$.
Let $\mathcal{T}^0$ denote the subscheme
of $\mathcal{T}$ whose geometric fibers are $\mathcal{T}_p^0$.
My question is can I express the finite index $[\mathcal{T}(A_\infty):\mathcal{T}^0(A_\infty)]$
in terms of the product of $[\mathcal{T}_p(\mathcal{O}_p):\mathcal{T}_p^0(\mathcal{O}_p)]$
(which may differ from 1 only if $p$ is ramified on the minimal splitting field of $T$ T$) ?)
Thank you, Rony.
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3
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Hi.
Let $K = F_q(C)$ be a global function field of an irreducible projective and smooth curve $C$
defined over a finite field of constants $F_q$. 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 $\mathcal{T}_p$
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 $\mathcal{T}_p^0$ the open subscheme
of $\mathcal{T}_p$ having a connected reduction modulo $p$.
Let $\mathcal{T}^0$ denote the subscheme
of $\mathcal{T}$ whose geometric fibers are $\mathcal{T}_p^0$.
My question is can I express the finite index $[\mathcal{T}(A_\infty):\mathcal{T}^0(A_\infty)]$
in terms of the product of $[\mathcal{T}_p(\mathcal{O}_p):\mathcal{T}_p^0(\mathcal{O}_p)]$
(which may differ from 1 only if $p$ is ramified on the minimal splitting field of $T$ ?)
Thank you, Rony.
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2
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1
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Group of connected components of the global N\'eron-Raynaud model of a torus
Hi.
Let $K = F_q(C)$ be a global function field of an irreducible projective and smooth curve $C$
defined over a finite field of constants $F_q$. 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 $\mathcal{T}_p$
which are of finite type,
defined each one over the corresponding local valuation ring $\mathcal{O}_p$.
Denote by $\mathcal{T}_p^0$ the open subscheme
of $\mathcal{T}_p$ having a connected reduction modulo $p$.
Let $\mathcal{T}^0$ denote the subscheme
of $\mathcal{T}$ whose geometric fibers are $\mathcal{T}_p^0$.
My question is can I express the finite index $[\mathcal{T}(A_\infty):\mathcal{T}^0(A_\infty)]$
in terms of the product of $[\mathcal{T}_p(\mathcal{O}_p):\mathcal{T}_p^0(\mathcal{O}_p)]$
(which may differ from 1 only if $p$ is ramified on the minimal splitting field of $T$ ?)
Thank you, Rony.
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