Open Problems on Central Simple Algebras — Conjecture 24 (Tate's Conjecture T)

Definitions and mathematical statement

Let $\ell\neq p$ be a prime and write \[ \overline X=X\times_k k_{\mathrm{sep}},\qquad V_\ell=\mathrm H^2(\overline X,{\mathbb{Q}}_\ell(1)), \] where $k_{\mathrm{sep}}$ denotes a separable closure of $k$, and ${\mathbb{Q}}_\ell(1)$ denotes the first Tate twist. The supplied record refers to surfaces, but does not specify the hypotheses on $X$ or $k$, define $p$ and $q$, or explicitly identify $k$ as a finite field of cardinality $q$ and characteristic $p$. These omissions must be resolved before this becomes a fully specified conjecture.

Following the supplied notation, let $\operatorname{Pic}(X)$ be the group of line bundles on $X$, let $\operatorname{Pic}^0(X)$ denote its subgroup of classes algebraically equivalent to zero, and set \[ \operatorname{NS}(X)=\operatorname{Pic}(X)/\operatorname{Pic}^0(X),\qquad \rho(X)=\operatorname{rank}_{{\mathbb{Z}}}\operatorname{NS}(X). \] The record uses $X$, rather than $\overline X$, in this definition; no replacement by the geometric Néron--Severi group is made here.

Let $F$ denote the action described in the record as geometric Frobenius on $V_\ell$. The record calls $P_2(X,t)$ its characteristic polynomial but supplies no determinant formula. In particular, the two polynomial conventions \[ \det(t\,\mathrm{id}_{V_\ell}-F) \quad\text{and}\quad \det(\mathrm{id}_{V_\ell}-tF) \] have different root conventions and must be distinguished.

Recorded assertion (Tate's Conjecture T, source Conjecture 24). With the source's unresolved hypotheses and normalization, the assertion is \[ \rho(X)=\operatorname{ord}_{t=q^{-1}}P_2(X,t), \] where $\operatorname{ord}_{t=a}P(t)$ is the multiplicity of $a$ as a root of a nonzero polynomial $P$. The simultaneous use of geometric Frobenius, the Tate twist ${\mathbb{Q}}_\ell(1)$, and the root $q^{-1}$ requires source verification: the record does not establish a compatible normalization. The displayed equality preserves the supplied formulation without silently changing the twist, Frobenius convention, or root.

Short English statement

The supplied version of Tate's Conjecture T equates the Picard number of a surface $X$ with the multiplicity of $q^{-1}$ in a Frobenius polynomial. The hypotheses on the surface and base field are missing, and the polynomial convention, Tate twist, and stated root require reconciliation before a precise conjecture can be finalized.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800712$, exact source problem code catalogue entry, AMR Open Problem Lists. The registry labels the problem open; that label is supplied metadata, not an independently verified literature assessment.
• Auel, Brussel, Garibaldi, Vishne, Open Problems on Central Simple Algebras (2010), Conjecture 24, identified in the record as Tate's Conjecture T. Supplied source URL: https://arxiv.org/abs/1006.3304. The record reports extraction from arXiv LaTeX and an access date of 2026-10-16; this is supplied provenance, not an independently performed source visit.
• Tate, reference label [Ta65], $\S 3$, as cited in the supplied statement. No complete bibliographic entry for this reference is supplied.