Finiteness of Frobenius Traces of a Sheaf on a Flat Arithmetic Scheme

Abstract

Abstract For a lisse $\ell $-adic sheaf on a scheme flat and of finite type over $\mathbb{Z}$, we consider the field generated over $ \mathbb{Q}$ by Frobenius traces of the sheaf at closed points. Assuming conjectural properties of geometric Galois representations of number fields and the Generalized Riemann Hypothesis, we prove that the field is finite over $\mathbb{Q}$ when the sheaf is de Rham at $\ell $ pointwise. This is a number field analog of Deligne’s finiteness result about Frobenius traces in the function field case.