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There are many analogies between $\ell$-adic sheaves on varieties over finite fields and vector bundles with connections on varieties over fields of characteristic zero. I would like to know what is the equivalent of the residue. Let me explain what I mean.

Let $(E, \nabla)$ be a vector bundle with connection on some smooth variety $X$ over a field $k$ of characteristic zero. Assume that $(E, \nabla)$ has regular singularities, meaning that there is a logarithmic extension $\bar{E} \to \bar{E} \otimes \Omega_{\bar{X}}(\log D)$, where $\bar{X}$ is a smooth compactification of $X$ such that $D=\bar{X}-X$ has normal crossings. Then, for each irreducible component of $D_i$, the composition with the Poincare residue gives a map $$ \bar{E} \to \bar{E} \otimes \mathcal{O}_{D_i} $$ which in fact restricts to an endomorphism of $\bar{E} \otimes \mathcal{O}_{D_i}$.

What is the $\ell$-adic analogue of this? Assume we are over a finite field and in the same situation as before ($X \hookrightarrow \bar{X}$ good compactification, this is not anymore automatic!). A regular singular connection should be replace by a representation $\pi_1^{et}(X) \to GL_n(\bar{\mathbb{Q}}_\ell)$ tamely ramified at $D$. How do I get the residue?

One of my concerns is that, in order to define the residue, I need to choose a logarithmic extension and the residue does depend on this choice. What is the substitute of the logarithmic extension in the $\ell$-adic context?

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    $\begingroup$ An extension of the sheaf to the Kummer \'etale site of $(X, D)$ (where we allow covers tamely ramified along $D$)? $\endgroup$ – Piotr Achinger Sep 30 '14 at 22:42
  • $\begingroup$ "singular regularities" or regular singularities? $\endgroup$ – abx Oct 1 '14 at 6:34
  • $\begingroup$ Sorry, it is corrected now. Do you have some deeper knowledge to share with us? ;) $\endgroup$ – glen90 Oct 1 '14 at 6:37
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So in the \'{e}tale case basically the only thing you have access to is the tame ramification group action on the nearby cycles sheaf on $D$. Because the tame ramification group is cyclic, this is basically the action of a matrix.

I guess the residue of the equation $dy/d \log x = \alpha$ is $\alpha$? So that equation corresponds to the matrix $e^{ 2\pi i \alpha}$. If this is right, then clearly you want to choose a logarithm of the matrix, and take that to be your residue map.

However you face a problem because most matrices don't have logarithms $\ell$-adiccally. So you could restrict attention to matrices that are unipotent modulo the appropriate power of $\ell$, or alternately you can base change from $\mathbb Q_\ell$ to $\mathbb C$ and take a logarithm there.

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  • $\begingroup$ Hi Will, thank's for your answer! I think that's a good guess. Can you say something else about how this tame ramification group action constructed or tell me some precise place where I can read about it? $\endgroup$ – glen90 Oct 4 '14 at 14:47
  • $\begingroup$ Abyankhar's lemma says that the action of the fundamental group on a tamely ramified torsion sheaf factors throughout the quotient coming from the cover where you adjoin the $n$th root of the equation defining the divisor. This group is cyclic. Then for an $\ell$-adic cover you get an action of the inverse limit group, which is infinite cyclic. $\endgroup$ – Will Sawin Oct 4 '14 at 19:00

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