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I'm currently reading Mar's and Springer's Character Sheaves. In Chapter 2 (Kummer local systems on tori), they provide a construction of Kummer local systems on a torus $T$ by way of the $m^{th}$ power isogeny, whose Galois covering $_m T$ is the group of elements of $T$ with order dividing $m$. For a character $x$ of $T$, we get a character $\chi_{x,m}$ of $_m T$ by $$\chi_{x,m}(t) = x(t).$$

They then remark that this lifts to a one-dimensional representation of $\pi_1(T, e)$ and so the equivalence of categories between local systems and $\pi_1(T,e)$-reps. by monodromy gives us a local system on $T$.

My question is: why does $\chi_{x,m}$ lift to a $\pi_1(T,e)$-rep. (what are we even lifting over?) and does the resulting local system have an easy explicit description?

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For anyone interested: it turns out that the fundamental group in question is the étale fundamental group, which is defined as a projective limit along the automorphism groups of a pro-representing system for the finite étale covering spaces of $T$. As such, $\pi_1(T,e)$ by definition comes equipped with a map to $_m T$ (since this is the automorphism group of the Galois covering $m$) and this is what we are lifting over.

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    $\begingroup$ (1) Note that for the usual fundamental group, the fact that $\pi_1(X)$ maps to the automorphism group of any Galois finite étale covering of $X$ is no longer true by definition, but it is still true, by a short argument. (2) In the future, people may be more likely to see your question if you use an arXiv tag (like ag.algebraic-geometry for this one). I would have answered it if I'd seen it! $\endgroup$
    – Will Sawin
    Commented Aug 26, 2021 at 1:13
  • $\begingroup$ @Will Sawin that’s good to know, thank you. I have another question that hasn’t gotten a lot of attention, even after I assigned a bounty, so I might try that. $\endgroup$
    – Martin Skilleter
    Commented Aug 26, 2021 at 2:44
  • $\begingroup$ @MartinSkilleter, don't forget to accept your own answer to get this off the unanswered questions list! $\endgroup$
    – LSpice
    Commented Aug 26, 2021 at 21:45

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