Frobenius trace fields of cohomologically rigid local systems

TL;DR

This paper studies the fields generated by Frobenius traces of cohomologically rigid local systems on complex varieties, showing under certain conditions these fields are uniformly bounded across primes, linking to geometric origin properties.

Contribution

It proves boundedness of Frobenius trace fields for cohomologically rigid local systems under various conditions, connecting arithmetic properties to geometric origin concepts.

Findings

Frobenius trace fields are bounded when local monodromy is totally degenerate.

Boundedness of Frobenius fields at one point implies boundedness everywhere.

Connections between boundedness and being strongly of geometric origin.

Abstract

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.