Common Branched Covers from Equal Hasse–Weil Zeta Functions

Definitions and mathematical statement

Let $X,X'$ be smooth projective varieties over ${\mathbb{Q}}$, and suppose their Hasse--Weil zeta functions, with the same local-factor conventions, satisfy \[ \zeta_X^{\mathrm{HW}}(s)=\zeta_{X'}^{\mathrm{HW}}(s). \] At a prime of good reduction, the usual local factor is obtained from \[ Z(X_p,T)=\exp\left(\sum_{r\geq1}\#X_p({\mathbb{F}}_{p^r})\frac{T^r}{r}\right), \] with $T=p^{-s}$; the treatment of bad primes and whether equality means a full or incomplete Euler product are not specified in the extract.

The source asks whether there is a common finite branched cover. In the literal algebraic reading, this means a variety $Z$ with finite surjective morphisms \[ Z\xrightarrow{\pi}X,\qquad Z\xrightarrow{\pi'}X', \] with ramification permitted. This is stronger than merely having two dominant rational maps from a resolution, but weaker than requiring both maps to be finite étale. The source does not specify whether $Z$ must be smooth, normal, geometrically integral, or defined over ${\mathbb{Q}}$ rather than a field extension, nor any restrictions on branching or degrees. Those choices are essential to identify the intended theorem; they are not fixed by the phrase alone. A literal unrestricted branched-cover question must also be distinguished from a more restrictive commensurability question, and no stronger unramified condition is inserted silently.

On a specified common-cover convention, the proposed implication is therefore \[ \forall X,X'\quad \zeta_X^{\mathrm{HW}}=\zeta_{X'}^{\mathrm{HW}} \ \Longrightarrow\ \exists(Z,\pi,\pi')\text{ satisfying that convention}. \] Connectedness and geometric-integrality conventions for the original varieties also require confirmation. A common finite cover by an integral $Z$ would in particular have the same dimension as both targets; this is not an extra equality-of-dimension hypothesis supplied by the text.

The supplied literature assessment checked 2026-08-21 reports that the abelian-variety case follows from Faltings' isogeny theorem, while supplying no general theorem for all smooth projective varieties. An isogeny gives a specific finite-cover relationship in that reported case. The report's general partially solved label is not independently certified for every interpretation of unrestricted branched cover.

Short English statement

Does equality of Hasse--Weil zeta functions force a common finite cover with allowed branching? The source must specify the cover's field, smoothness, and ramification requirements; replacing branched by étale would change the question.

Sources

• Local UnsolvedMath registry, permanent ID 30000405, catalogue entry, Common Branched Covers from Equal Hasse--Weil Zeta Functions.
• The Arithmetic of Fields (2006), Oberwolfach Reports, pp. 317--378, DOI https://doi.org/10.4171/owr/2006/06.
• The supplied assessment attributes the abelian-variety case to Faltings' isogeny theorem but gives no separate bibliographic citation or exact theorem statement.
• Supplied literature triage checked 2026-08-21; it cites only the original report and supplies no precise general common-cover convention.