Bogomolov property for rational-map moduli outside the flexible Lattès locus

Establish that for every number field K and every integer d≥2, there exists ε_{K,d}>0 such that the set of points α in the degree-d rational-map moduli space M_d(K^{ab}) outside the flexible Lattès locus [FL_d] with critical height h_crit(α)<ε_{K,d} equals the finite zero-critical-height locus outside [FL_d].

Background

The paper proves an analogous strong Bogomolov property for the critical height on the moduli space MPolyd of degree-d polynomials over maximal abelian extensions. For general rational maps, the flexible Lattès locus must be removed because, when d is a square, it contains infinitely many classes defined over the rationals, all of which are postcritically finite and have critical height zero.

The conjecture asks for a uniform positive lower bound on positive critical heights of rational-map classes over K{ab} outside this exceptional locus, together with finiteness of the zero-height classes there. It is the strongest of the three rational-map conjectures proposed in the paper.

References

Motivated by Theorems \ref{thm:main}, \ref{thm:pcf}, and \ref{thm:cyc}, we propose the following conjectures. \begin{conjecture}\label{conj:rbog} For every number field $K$ and every $d\geq2$, there exists $\epsilon_{K,d}>0$ such that

\left\lbrace\alpha\inM_d()\setminus[FL_d]\colon h_{\mathrm{crit}(\alpha)<\epsilon_{K,d}\right\rbrace=\left\lbrace\alpha\inM_d()\setminus[FL_d]\colon h_{\mathrm{crit}(\alpha)=0\right\rbrace

is finite. \end{conjecture}

— A Bogomolov property for moduli spaces of polynomials over abelian extensions  (2610.06382 - Zhang, 5 Oct 2026) in Section 1, subsection “Rational map conjectures,” Conjecture 1 (labelled conj:rbog)

\begin{conjecture}\label{conj:rab} For every number field $K$ and every $d\geq2$, the set

R_d():=\left\lbrace[f]\inM_d()\setminus[FL_d]\colon f\text{ is PCF}\right\rbrace

is finite. \end{conjecture}

— A Bogomolov property for moduli spaces of polynomials over abelian extensions  (2610.06382 - Zhang, 5 Oct 2026) in Section 1, subsection “Rational map conjectures,” Conjecture 2 (labelled conj:rab)

\begin{conjecture}\label{conj:rcyc} For every number field $K$ and every $d\geq2$, the set

R_d():=\left\lbrace[f]\inM_d()\setminus[FL_d]\colon f\text{ is PCF}\right\rbrace

is finite. \end{conjecture}

— A Bogomolov property for moduli spaces of polynomials over abelian extensions  (2610.06382 - Zhang, 5 Oct 2026) in Section 1, subsection “Rational map conjectures,” Conjecture 3 (labelled conj:rcyc)