  Let $X/\mathbb{C}$ be a smooth variety with simple normal crossings
compactification $\bar{X}$, and let $L$ be an irreducible
$\overline{\mathbb{Q}}_{\ell}$-local system on $X$ with torsion determinant.
Suppose $L$ is cohomologically rigid. The pair $(X, L)$ may be spread out to a
finitely generated base, and therefore reduced modulo $p$ for almost all $p$;
the Frobenius traces of this mod $p$ reduction lie in a number field $F_p$, by
a theorem of Deligne. We investigate to what extent the fields $F_p$ are
bounded, meaning that they are contained in a fixed number field, independent
of $p$. We prove a host of results around this question. For instance: assuming
$L$ has totally degenerate unipotent monodromy around some component of $Z$,
then we prove that $L$ admits a spreading out such that the $F_p$'s are
bounded; without any local monodromy assumptions, we show that the $F_p$'s are
bounded as soon as they are bounded at one point of $X$.
  We also speculate on the relation between the boundedness of the $F_p$'s, and
the local system $L$ being strongly of geometric origin, a notion due to
Langer-Simpson.
