Generation of trace fields of Frobenii on local systems

Question

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 in Section 0.1 of the linked paper.

Answer 1

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 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.