Andrews–Petsche conjecture on abelian backward orbits

Let $K$ be a number field, let $f\in K(x)$ be a rational map of degree at least $2$ , and let $\alpha\in\mathbb{P}^{1}(\overline{K})$ be non-exceptional, meaning that its backward orbit $f^{-\infty}(\alpha)=\{\beta\in\mathbb{P}^{1}(\overline{K}):f^{n}(\beta)=\alpha\text{ for some }n\ge 0\}$ is infinite. Set $L=K(f^{-\infty}(\alpha))$ . The Andrews–Petsche conjecture asserts that $\operatorname{Gal}(L/K)$ is virtually abelian if and only if $\alpha$ is $f$ -preperiodic and $f$ is an exceptional map with complex multiplication; in the one-dimensional powering and Chebyshev formulations, this includes the cases obtained from $x\mapsto x^{d}$ and $x\mapsto \pm T_{d}(x)$ by conjugation.

References

Primary source

arXiv

Additional references

Exceptional maps and abelian points in backward orbits — arXiv — Zhuchao Ji, Jiarui Song, Junyi Xie

algebraic-geometry arithmetic-dynamics

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A September 2026 paper claims to prove the conjecture and extend its classification from rational maps to higher-dimensional varieties, but the result has not been independently verified.

The Andrews–Petsche conjecture predicts that virtually abelian backward-orbit Galois groups arise exactly from powering and Chebyshev-type dynamics, subject to a non-exceptional-point hypothesis. Earlier literature recorded only partial results.

Known results

Sufficiency was proved for powering and Chebyshev cases, including the corrected $\pm T_d$ formulation.
The conjecture was proved for stable quadratic postcritically infinite maps over $\mathbb{Q}$ .
Ferraguti and Pagano (2023) proved it for unicritical polynomials with periodic critical orbit and for all monic unicritical polynomials over quadratic fields.
They also bounded exceptional basepoints outside a finite degree-dependent set.

September 2026 higher-dimensional extension

Zhuchao Ji, Jiarui Song, and Junyi Xie report in Exceptional maps and abelian points in backward orbits that the conjectural characterization is proved and extended from rational maps to higher-dimensional normal projective varieties. This is a complete-resolution claim, but the retrieved material provides no independent verification.

Current status (as of September 2026): The conjecture is claimed proved, including a higher-dimensional extension, but that claim remains unverified; the non-exceptional-point assumption remains part of the stated theorem.

Sources

No solutions have been posted yet.