Let $X$ be an (as nice as you prefer) alg. variety (or alg. stack) defined over $\mathbb{F}\_{q}$ and let $\mathcal{F}$ be an l-adic sheaf on $X\_n = X {\times\_{\mathbb{F}q}} \mathbb{F}\_{q^n}$. Fix an isomorphism between $\mathbb{C}$ and $\overline{\mathbb{Q}}\_l$ (otw replace below $\mathbb{C}$ by the $l$-adic numbers).
Suppose we know that the sheaf $\mathcal{F}$ comes (by pullback) from a sheaf, say $\mathcal{G}$, defined on $X$.


I would like to know if it is possible to recover the trace function of $\mathcal{G}$, say $g = f\_\mathcal{G}:X(\mathbb{F}\_q) \to \mathbb{C}$ defined by 


$f\_\mathcal{G}(x) = tr(Frob\_x,\mathcal{G}\_x)$ 
from the (analogously defined) trace function of $\mathcal{F}$, say


$f:=f\_\mathcal{F}:X(\mathbb{F}\_{q^n})\to\mathbb{C}$


I see that already for $X=\mathbb{F}\_q$ you can only recover the trace function $g$
up to an $n$-th root of unity. So the following questions might look a bit silly:


1. What's the best thing that we can do in general? i.e. if I have $f$ can I find $g$ up to some constant ($n$-th root of unity; need properness?).
2. If I have $f$ what additional structure/information would I need to recover (uniquely) the function $g$?


Also, if you know any reference in which similar problems are treated please let me know.



recover trace of l-adic sheaves defined over an extension
As anon points out, the answer in general is that one cannot recover it up to trace. Indeed, a sheaf on $\mathbb F\_q$ is just a matrix up to conjugation. The corresponding sheaf on $\mathbb F\_{q^n}$ is just that matrix to the $n$th power. There is no relation between the traces.


However, if you have the characteristic polynomial of the matrix, you're in luck. The eigenvalues of the sheaf on $\mathbb F\_q$ must of course be $n$th roots of the eigenvalues of teh sheaf on $\mathbb F\_q^n$, which gives you at most $n^k$ possibilities for a sheaf of rank $k$.


Similarly, if we want to get down to a list of possibilities for a sheaf on a larger space, we need to look at the whole sheaf. So a different version of your question would be this:



> 
> Given a sheaf $\mathcal G$ on $X\_n$, how many descents does it have to $X$, and what are they?
> 
> 
> 


Then we compute the trace on each descent. A descent is just going to be a map $\mathcal G \to \left(Frob\_q\right)^\*\mathcal G$ such that the composition of the map with itself $n$ times is the identity.


One case in which you are alright is if $\mathcal G$ is lisse and irreducible, and the base is connected, so the automorphism group of $\mathcal G$ is just $\mathbb G\_m$. Then any two descents must differ by an $n$th root of unity, and their traces differ by an $n$th root of unity, and you're off.


If $\mathcal G$ is lisse and semisimple, then you can write it as a sum of irreducible pieces of this form. You can throw away the pieces that are not fixed by $Frob\_q$, because the trace on those parts will always be zero. Then, on a piece that is fixed, the automorphism group will be $\mathbb G\_m$, so the trace will be defined by the sheaf up to a $n$th root of unity.


If you are not lisse or not semisimple, things get a bit messier.


