Ledet’s conjecture on the essential dimension of cyclic p-groups

Ledet’s conjecture asserts that over a field of characteristic $p>0$, the essential dimension of the cyclic group $C_{p^n}$ equals $n$. It is known in the first two cases, while the cases $n\ge 3$ remain unresolved.

Known results

• The upper bound $\operatorname{ed}_k(C_{p^n})\le n$ holds for all $n\ge 1$.
• The conjecture is proved for $n=1$ and $n=2$.
• For $n\ge 2$, the general lower bound $\operatorname{ed}_k(C_{p^n})\ge 2$ is known.
• Essential dimension at $p$ satisfies $\operatorname{ed}_k(C_{p^n};p)=1$, but this does not settle ordinary essential dimension.

October 2026 finite-degree profile results

Abhishek Shukla’s paper determines exact first-jump degrees, gives a prime-to-$p$ upper bound, and determines the $(p,n)=(2,2)$ profile. The $(2,3)$ slot remains unresolved, so this is claimed progress rather than a resolution.

Current status (as of October 2026): Ledet’s conjecture is settled for $n=1,2$ and remains open for $n\ge 3$, including the explicitly unresolved $(p,n)=(2,3)$ case.