I am trying to understand a certain sentence in a paper that I am reading. Let me start with some notation/background. (For a motivation of why this should be interesting, see below, under the questions.)


Notation/background
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First the (pretty standard) setup:


* $k$ a field with a discrete valuation $v$;
* $\mathcal{O}\_{v}$ the ring of integers;
* $\kappa$ the residue field (we assume it is finite);
* $\kappa\_{\text{s}}$ a seperable closure of $\kappa$;
* $G\_{\kappa}$ the absolute Galois group of $\kappa$;
* $F$ the Frobenius generator *($x \to x^{|\kappa|}$ on sections)*;
* $X/k$ a curve with semistable model $\mathcal{X} \to \operatorname{Spec} \mathcal{O}\_{v}$.


We can look at the Jacobian of $X$, which is an abelian variety $\operatorname{Jac} X$ over $k$. By the theory of Ńeron models we can form a smooth model $\mathcal{J}$ over $\operatorname{Spec} \mathcal{O}\_{v}$. Let $\tilde{J}$ denote the reduction at $v$, i.e., $\mathcal{J} \times\_{\mathcal{O}\_{v}} \kappa$. This is a commutative group scheme, and the component of the identity, $\tilde{J}^{0}$ fits is the extension of an abelian variety $A/\kappa$ by a linear group.


*Minor question: If I am not mistaken, this linear group is a torus $T/\kappa$, because our curve $X$ has a semistable model. Is this correct?*


Thus we have an exact sequence of commutative $\kappa$-group schemes:
`\[ 1 \to T \to \tilde{J}^{0} \to A \to 0. \]`


The paper that I am reading now considers



> 
> [...] $\tau = \pm 1$, the determinant of the action of $F$ on the character group of $T$.
> 
> 
> 


I searched the literature and the interwebs to get a hang of what is going on here, but I am not really confident of what I found. (Especially because I do not get $\tau = \pm 1$.)




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My guess
--------


According to <http://www.encyclopediaofmath.org/index.php/Character_group> the character group of $T$ is $\operatorname{Hom}(T, \mathbb{G}\_{\text{m}})$, i.e., $\mathbb{G}\_{\text{m}}(T)$. However, I could not think of any Galois action on this. I proceeded by guessing that $X(T) = \operatorname{Hom}(T(\kappa\_{\text{s}}), \mathbb{G}\_{\text{m}}(\kappa\_{\text{s}}))$ would be a good candidate for this character group, and moreover it carries a natural action of $G\_{\kappa}$ given by $f \cdot \sigma = f \circ \sigma$.


Let $e$ denote the dimension of $T$. Then we have the identities
`\[ X(T) = \operatorname{Hom}((\kappa_{\text{s}}^{*})^{e}, \kappa_{\text{s}}^{*}) = \operatorname{Hom}(\kappa_{\text{s}}^{*}, \kappa_{\text{s}}^{*})^{e}. \]`
(By definition of algebraic torus and the universal property of direct sums.)


Now I wanted to understand the determinant of $F$ acting on $X(T)$. It seemed natural to me to view $X(T)$ as free module of rank $e$ over $R = \operatorname{End}(\kappa\_{\text{s}}^{\*})$. The action of $F$ would then be given by the scalar matrix $|\kappa| \cdot I$. Its determinant would then be $|\kappa|^{e}$. Unless $e = 0$ (in the case of good reduction) this is not equal to $\pm 1$.


Likely I am messing things up horribly. First of all my computation of $\tau$ is not equal to $\pm 1$, and secondly it seems to depend only on $e$. (I guess it should be more intricately connected to $T$ as $k$-scheme, instead of only $T\_{\kappa\_{\text{s}}}$, the base change to the seperable closure.)




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Question
========



> 
> 1. Where did I take the wrong turn?
> 2. What is the right way of computing $\tau$.
> 
> 
> 




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Motivation
----------


Given the computation of $\tau$, we can 'easily' compute a certain *local root number* $\epsilon\_{v}$. This local root number is a local factor in the sign $\epsilon$ of the (conjectured) functional equation of the $L$-function of a certain motive $M$ associated to $X$.


The Beilinson-Bloch conjecture link the order of vanishing (at a certain critical point) of this $L$-function to the rank of the Chow group of $M$. Under certain conditions on $X$, one can construct a non-trivial element $\Delta\_{\xi}$ of $\operatorname{Ch}(M)$, hence proving that its rank is strictly positive. Assuming the truth of this conjecture, if $\epsilon = 1$, it follows that the rank is at least $2$.


And yep, that is why I think it is interesting to compute $\tau$.


For more information I refer to section 5 of Shou-Wu Zhang's paper *“Gross–Schoen Cycles and Dualising Sheaves”*, available at <http://arxiv.org/abs/0812.0371> .




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Edits
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As noted in my comment below. I stupidly overlooked the fact that $R$ is not a commutative ring.


Further I also found <http://www.martinorr.name/blog/2010/01/24/character-groups-of-algebraic-tori> which is really helpful. I have not fully figured out how to compute $\tau$. But at least it points in a different direction than my guess. (And I think the new direction is more promising.)



Understanding the determinant of the action of Frobenius on the character group of the toric part of the reduction of the Jacobian of a curve.
You need to work with just regular homomorphisms between those two groups, i.e. algebraic characters. That does two things.


First, note that $\operatorname{End}(\kappa\_s^\*)=\mathbb Z$.


Second, $f \circ \sigma$ is not in general a regular homomorphism. You need to take $\sigma^{-1} \circ f \circ \sigma$.


The key point is that this action on $X(T)$ is invertible, so lies in $GL\_n(\mathbb Z)$, so has determinant $\pm 1$.


