Meromorphic continuation of Hasse–Weil zeta functions for finite-type schemes over Z

Determine whether the Hasse–Weil zeta function ζ(X,s) attached to an arbitrary finite-type Z-scheme X admits analytic or meromorphic continuation to the complex plane, and establish such a continuation in general.

Background

The paper recalls known cases where meromorphic continuation is established (finite fields, smooth varieties over finite fields via the Weil conjectures, Dedekind zeta functions of number fields), and notes that beyond these settings the question remains unresolved.

A general meromorphic continuation for Hasse–Weil zeta functions would unify analytic properties across arithmetic and geometric contexts and is a long-standing open problem in arithmetic geometry.