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.

2012-11-07T07:36:41Z

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 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

Where did I take the wrong turn?

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 .

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.)

Answer by Will Sawin for 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.

2012-11-07T15:51:01Z

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 $\pm 1$.