Let k be a field of dimension d, and let b1∈operatornameBr(k). The index ind(alpha) is the square root of the rank of the unique division algebra representing alpha. Period-index conjecture. ind(alpha)| (per(alpha))^d-1 for all alpha∈operatornameBr(k). This is the algebraic period-index conjecture attributed in the source to Colliot-Thélène. The surrounding text presents a topological period-index theorem as evidence, but gives no resolution of this conjecture.

Colliot-Thélène's period-index conjecture

About 17 years old · traced to 2009

Let $k$ be a field of dimension $d$ , and let $b1\in\operatorname{Br}(k)$ . The index $ind(\alpha)$ is the square root of the rank of the unique division algebra representing $\alpha$ .

Period-index conjecture.

ind(\alpha)\mid (per(\alpha))^{d-1}

for all $\alpha\in\operatorname{Br}(k)$ .

This is the algebraic period-index conjecture attributed in the source to Colliot-Thélène. The surrounding text presents a topological period-index theorem as evidence, but gives no resolution of this conjecture.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Colliot-Thélène's period-index conjecture
Let $X$ be a smooth projective variety over an algebraically closed field, and let $\alpha\in\operatorname{Br}(X)$ be a Brauer class. Write $\operatorname{per}(\alpha)$ for its period and $\operatorname{ind}(\alpha)$ for its index. Colliot-Thélène's conjecture. Every such class satisfies
$\operatorname{ind}(\alpha)\mid\operatorname{per}(\alpha)^{\dim(X)-1}.$
This is the general period-index conjecture. The conjecture is wide open in general, although several cases and partial bounds are known.
source: Alessio Bottini and Daniel Huybrechts, “The period-index problem for hyperkähler varieties: Lower and upper bounds”, arXiv:2512.15131 (2025).

References

Primary source

Benjamin Antieau, “Cohomological obstruction theory for Brauer classes and the period-index problem”, arXiv:0909.2352 (2010).

Posted 1 month ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A 2026 preprint claims the conjecture is false in three or more dimensions, but its counterexamples have not been independently verified.

Attributed to Colliot-Thélène, the conjecture predicts that every Brauer class over a field of dimension $d$ has index dividing its period raised to the power $d-1$ . A 2026 preprint claims explicit counterexamples beginning in dimension $3$ .

Known results

The conjecture is vacuous for $d\leq 1$ .
It holds for $d=2$ by work of de Jong, Lieblich, and de Jong–Starr.
A 2024 paper proves $\operatorname{ind}(\alpha)\mid\operatorname{per}(\alpha)^2$ for suitable classes on abelian threefolds, not the general case.
Topological and Hodge-theoretic results give evidence or conditional bounds, but do not resolve the algebraic conjecture.

August 2026 counterexample claim

The preprint The period-index conjecture is false claims, over every algebraically closed characteristic-zero field, a smooth projective threefold with $\operatorname{per}(\alpha)=2$ and $\operatorname{ind}(\alpha)=8$ , and higher-dimensional examples with $\operatorname{ind}(\alpha_d)=2^d$ . It reports that ChatGPT assisted the author’s discovery process; the author wrote the paper and accepts responsibility for it.

Current status (as of September 2026): The cases $d\leq 2$ are settled, while the claimed counterexamples for $d\geq 3$ remain unverified.

Sources

No solutions have been posted yet.