2 Fixed a typo. Added new thoughts.

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

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 $\tau = \pm 1$.)

## My guess

According to [http://www.encyclopediaofmath.org/index.php/Character_group][1] 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 $R = \operatorname{End}(\kappa_{\text{s}})$. 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.) # Question 1. Where did I take the wrong turn? 2. What is the right way of computing$\tau$. ## 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][2] http://arxiv.org/abs/0812.0371 . ## Edits 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.) 1 # 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. 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 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$\tau = \pm 1$.) ## My guess According to [http://www.encyclopediaofmath.org/index.php/Character_group][1] 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.) # Question 1. Where did I take the wrong turn? 2. What is the right way of computing$\tau$. ## 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][2] .