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.

References

The existence of analytic/meromorphic continuation of the Hasse-Weil $\zeta$-functions for finite type varieties over $\mathbb{Z}$ is open in general.

— Equivariant algebraic $\mathrm{K}$-theory and Artin $L$-functions  (2405.03578 - Elmanto et al., 2024) in Remark, Section 2.1 (Zeta functions)

For any number field $k$, it is conjectured that $L(E/k,s)$ has analytic continuation to the whole complex plane. In the known cases, the analytic continuation is usually proved by converting this $L$-function into an analytic $L$-function, such as the $L$-function of a Hecke character or of a modular form, and using the analytic continuation of the latter. For general number fields, this conjecture remains open.

— A $p$-part Birch and Swinnerton-Dyer Formula over Totally Imaginary Quadratic Extension of Totally Real Fields  (2608.11969 - Li, 12 Aug 2026) in Remark following the definition of the Hasse–Weil L-function, Section 2.2.3