Let $\overline{X}$ be a smooth proper curve over $\mathbb{F}\_q$, for some $q$, $S$ a collection of $\mathbb{F}\_q$ points of $\overline{X}$, and set $X=\overline{X}-S$.


For a rank $n$ $\overline{\mathbb{Q}}\_{\ell}$-local system on $X$, it is known that the coefficients of the characteristic polynomials of Frobenii at all closed points generate a number field $E$, by the work of L. Lafforgue on function field Langlands.


**Question**: are there (known or conjectural) bounds on how many closed points are needed so that the coefficients of the Froebnii characteristic polynomials at these points generate the field $E$?


For example, it seems to me that in the case of $\overline{X}=\mathbb{P}^1\_{\mathbb{F}\_q}$, $S$ being four points, and $n=2$, the Frobenii at the $\mathbb{F}\_q$-points suffice to generate $E$, by the [computations of Kontsevich](https://www.math.nyu.edu/%7Etschinke/.manin/final/kontsevich/kontsevich.pdf) in Section 0.1 of the linked paper.



Generation of trace fields of Frobenii on local systems
Your proof in the case of four points assumes that the local monodromies at those four points are unipotent.


In general, one needs a bound not just on the set of ramification points but on the breaks/slopes of the sheaf at those points.


To see this is necessary, one can work already in the case of sheaves lisse of rank one on $\mathbb P^1\_{\mathbb F\_p}$ minus one point, where the Artin-Schreier sheaf $\mathcal L\_\psi ( x (x^{p^n}-x))$ has trace of Frobenius 1 at every point of degree dividing $n$ but has trace field $\mathbb Q(\mu\_p)$.


With this caveat, a bound is given by Deligne in Proposition 2.10 of the article [Finitude de l’extension de $\mathbb Q$ engendrée par des traces de Frobenius, en caractéristique finie](https://publications.ias.edu/sites/default/files/FrobTraces.pdf) which states that it suffices to take all closed points of degree at most


$$2n + 2 \log\_q^+ ( 2n^2 (b\_1 ( X) + \sum\_{s\in S} \alpha\_s \deg s))$$
where $\alpha\_s$ is the largest slope of the rank $n$ local system at $s$, $b\_1(X)$ is the first Betti number, and $\log\_q^+$ denotes the log base $q$, or $0$, whichever is larger.


