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. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T14:39:22Z http://mathoverflow.net/feeds/question/111712 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111712/understanding-the-determinant-of-the-action-of-frobenius-on-the-character-group-o 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. Johan Commelin 2012-11-07T07:36:41Z 2012-11-07T15:51:01Z <p>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.)</p> <h2>Notation/background</h2> <p>First the (pretty standard) setup:</p> <ul> <li>$k$ a field with a discrete valuation $v$;</li> <li>$\mathcal{O}_{v}$ the ring of integers;</li> <li>$\kappa$ the residue field (we assume it is finite);</li> <li>$\kappa_{\text{s}}$ a seperable closure of $\kappa$;</li> <li>$G_{\kappa}$ the absolute Galois group of $\kappa$;</li> <li>$F$ the Frobenius generator <em>($x \to x^{|\kappa|}$ on sections)</em>;</li> <li>$X/k$ a curve with semistable model $\mathcal{X} \to \operatorname{Spec} \mathcal{O}_{v}$.</li> </ul> <p>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., <code>$\mathcal{J} \times_{\mathcal{O}_{v}} \kappa$</code>. 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.</p> <p><em>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?</em></p> <p>Thus we have an exact sequence of commutative $\kappa$-group schemes: <code>$1 \to T \to \tilde{J}^{0} \to A \to 0.$</code></p> <p>The paper that I am reading now considers</p> <blockquote> <p>[...] $\tau = \pm 1$, the determinant of the action of $F$ on the character group of $T$.</p> </blockquote> <p>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 $\tau = \pm 1$.)</p> <hr> <h2>My guess</h2> <p>According to <a href="http://www.encyclopediaofmath.org/index.php/Character_group" rel="nofollow">http://www.encyclopediaofmath.org/index.php/Character_group</a> the character group of $T$ is <code>$\operatorname{Hom}(T, \mathbb{G}_{\text{m}})$</code>, i.e., <code>$\mathbb{G}_{\text{m}}(T)$</code>. However, I could not think of any Galois action on this. I proceeded by guessing that <code>$X(T) = \operatorname{Hom}(T(\kappa_{\text{s}}), \mathbb{G}_{\text{m}}(\kappa_{\text{s}}))$</code> would be a good candidate for this character group, and moreover it carries a natural action of $G_{\kappa}$ given by $f \cdot \sigma = f \circ \sigma$.</p> <p>Let $e$ denote the dimension of $T$. Then we have the identities <code>$X(T) = \operatorname{Hom}((\kappa_{\text{s}}^{*})^{e}, \kappa_{\text{s}}^{*}) = \operatorname{Hom}(\kappa_{\text{s}}^{*}, \kappa_{\text{s}}^{*})^{e}.$</code> (By definition of algebraic torus and the universal property of direct sums.)</p> <p>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 $\pm 1$.</p> <p>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 <code>$T_{\kappa_{\text{s}}}$</code>, the base change to the seperable closure.)</p> <hr> <h1>Question</h1> <blockquote> <ol> <li>Where did I take the wrong turn?</li> <li>What is the right way of computing $\tau$.</li> </ol> </blockquote> <hr> <h2>Motivation</h2> <p>Given the computation of $\tau$, we can 'easily' compute a certain <em>local root number</em> $\epsilon_{v}$. 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 $X$.</p> <p>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 $2$.</p> <p>And yep, that is why I think it is interesting to compute $\tau$.</p> <p>For more information I refer to section 5 of Shou-Wu Zhang's paper <em>“Gross–Schoen Cycles and Dualising Sheaves”</em>, available at <a href="http://arxiv.org/abs/0812.0371" rel="nofollow">http://arxiv.org/abs/0812.0371</a> .</p> <hr> <h2>Edits</h2> <p>As noted in my comment below. I stupidly overlooked the fact that $R$ is not a commutative ring.</p> <p>Further I also found <a href="http://www.martinorr.name/blog/2010/01/24/character-groups-of-algebraic-tori" rel="nofollow">http://www.martinorr.name/blog/2010/01/24/character-groups-of-algebraic-tori</a> 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.)</p> http://mathoverflow.net/questions/111712/understanding-the-determinant-of-the-action-of-frobenius-on-the-character-group-o/111735#111735 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. Will Sawin 2012-11-07T15:51:01Z 2012-11-07T15:51:01Z <p>You need to work with just regular homomorphisms between those two groups, i.e. algebraic characters. That does two things.</p> <p>First, note that $\operatorname{End}(\kappa_s^*)=\mathbb Z$.</p> <p>Second, $f \circ \sigma$ is not in general a regular homomorphism. You need to take $\sigma^{-1} \circ f \circ \sigma$.</p> <p>The key point is that this action on $X(T)$ is invertible, so lies in $GL_n(\mathbb Z)$, so has determinant $\pm 1$.</p>