On the rationality of algebraic monodromy groups of compatible systems

TL;DR

This paper proves the existence of a common rational form for algebraic monodromy groups of compatible systems over number fields, with applications to Mumford-Tate conjecture predictions.

Contribution

It constructs a common $E$-form of monodromy groups and tautological representations for compatible systems, extending rationality results in characteristic $p$ and zero.

Findings

Constructed a common $E$-form of monodromy groups $G$ for compatible systems.

Established rationality of monodromy groups assuming crystalline companions or ordinariness.

Applied results to construct compatible systems and support Mumford-Tate conjecture predictions.

Abstract

Let $E$ be a number field and $X$ a smooth geometrically connected variety defined over a characteristic $p$ finite field. Given an $n$-dimensional pure $E$-compatible system of semisimple $\lambda$-adic representations of the \'etale fundamental group of $X$ with connected algebraic monodromy groups $G_\lambda$, we construct a common $E$-form $G$ of all the groups $G_\lambda$ and in the absolutely irreducible case, a common $E$-form $G\hookrightarrow\text{GL}_{n,E}$ of all the tautological representations $G_\lambda\hookrightarrow\text{GL}_{n,E_\lambda}$ (Theorem 1.1). Analogous rationality results in characteristic $p$ assuming the existence of crystalline companions in $\text{F-Isoc}^{\dagger}(X)\otimes E_{v}$ for all $v|p$ (Theorem 1.5) and in characteristic zero assuming ordinariness (Theorem 1.6) are also obtained. Applications include a construction of $G$-compatible system from some $\text{GL}_n$-compatible system and some results predicted by the Mumford-Tate conjecture.