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?



Character constructed from Kummer local system lifts to representation of algebraic torus
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.


