Characterizing pseudo-Anosov measured foliations whose endomorphism-field extension is a ring class field (Nikolaev's measured-foliation approach to Hilbert's 12th problem)

To a pseudo-Anosov measured foliation $\mathcal{F}$ on a compact surface, Nikolaev's paper "Measured foliations and Hilbert 12th problem" (Acta Math. Vietnam. 48 (2023), 271-281) attaches two number fields: the endomorphism field $K=\mathbb{Q}\otimes\mathrm{End}(\mathrm{Jac}(\mathcal{F}))$ of the foliation's period (jacobian) module, and the real quadratic or rational subfield $k=\mathbb{Q}\otimes\mathrm{End}(\mathrm{Jac}(h(\mathcal{F})))$ obtained from an induced foliation on the torus. For foliations defined by maximal Hecke eigenforms on modular curves $X_0(N)$, the paper claims $\mathrm{Gal}(K|k)\cong\mathrm{Cl}(R)$, the ideal class group of the order $R$, i.e. that $K$ is a ring class field of $k$; its Remark 3.4 concedes that this phenomenon is not universal, since for a general pseudo-Anosov foliation $K|k$ need not be Galois, and a Galois $K|k$ need not have Galois group described by the class group. The characterization problem - determining precisely which geometric or arithmetic conditions on $\mathcal{F}$ force the class-field-theoretic isomorphism - remains open. A secondary, auxiliary gap is the missing conductor $\mathrm{N}(\mathfrak{c})$ for Shimura's level $N=233$ example.

Background

Hilbert's twelfth problem asks for explicit constructions of the abelian extensions of a number field, generalizing the Kronecker-Weber theorem (every finite abelian extension of $\mathbb{Q}$ lies in a cyclotomic field) and the theory of complex multiplication, which generates the abelian extensions of an imaginary quadratic field from special values of modular functions such as the elliptic $j$-invariant. For real quadratic fields no analogue is known: class field theory guarantees that every finite abelian extension of a real quadratic field $k$ sits inside ray- and ring-class fields - the abelian extensions whose Galois groups over $k$ are the ideal class groups $\mathrm{Cl}(R)$ of orders $R=\mathbb{Z}+\mathfrak{f}\mathcal{O}_k$ (subrings of the ring of integers determined by a conductor $\mathfrak{f}\geq 1$; $\mathfrak{f}=1$ gives the Hilbert class field) - but producing explicit generators of these fields from analytic special values remains open. Manin's program of real multiplication proposed replacing the period lattices of elliptic curves with complex multiplication by "pseudo-lattices" $\mathbb{Z}+\mathbb{Z}\theta\subset\mathbb{R}$ with $\theta$ real quadratic, whose endomorphism ring is exactly such an order $R$.

Nikolaev's paper develops this idea through surface dynamics. A measured foliation $\mathcal{F}$ on a compact surface $X$ is a singular foliation carrying a transverse measure, given locally by trajectories of a closed real 1-form (Hubbard and Masur, 1979); its jacobian $\mathrm{Jac}(\mathcal{F})$ is the $\mathbb{Z}$-module of periods of that form, a rank-$n$ subgroup of $\mathbb{R}$ ($n=2g+|\mathrm{Sing}\,\mathcal{F}|-1$ for genus $g$). By the Nielsen-Thurston classification, a surface automorphism of infinite order that preserves no simple closed curve is pseudo-Anosov: it stretches an invariant measured foliation by a real algebraic number $\lambda_\varphi>1$, the dilatation, and the periods of the invariant foliation lie in the number field $\mathbb{Q}(\lambda_\varphi)$ (Thurston, 1988). The paper introduces a map $h$ assigning to $\mathcal{F}$ a measured foliation $h(\mathcal{F})$ on the two-torus (via a branched-covering construction) with period-module inclusion $\mathrm{Jac}(h(\mathcal{F}))\subseteq\mathrm{Jac}(\mathcal{F})$, so that $k=\mathbb{Q}\otimes\mathrm{End}(\mathrm{Jac}(h(\mathcal{F})))$ is real quadratic or $\mathbb{Q}$, and $K=\mathbb{Q}\otimes\mathrm{End}(\mathrm{Jac}(\mathcal{F}))$ is a number field containing it.

For the modular case, $X=X_0(N)$ is the modular curve (the compactified quotient of the upper half-plane by the congruence subgroup $\Gamma_0(N)$ of $\mathrm{SL}_2(\mathbb{Z})$) and $\mathcal{F}=\mathcal{F}_N$ is the foliation by vertical trajectories $\Re(f\,dz)=0$ of a maximal Hecke eigenform $f$ - a weight-two cusp form that is a simultaneous eigenvector for all Hecke operators $T_n$ and whose Fourier coefficient field $K_f$ has degree equal to the genus $g$ of $X_0(N)$. The paper shows $\mathrm{Jac}(h(\mathcal{F}_N))$ is a Manin pseudo-lattice $\Lambda_M$ with order $R$, defines a $j$-invariant $j(\Lambda_M)$ as the associated Hecke unit (the Perron-Frobenius eigenvalue of the Hecke action on the period module), and asserts in its Theorem 1.1 that $K=k(j(\Lambda_M))$ is a ring class field of $k$ modulo $\mathfrak{f}$, i.e. $\mathrm{Gal}(K|k)\cong\mathrm{Cl}(R)$. The argument matches the $g$ conjugate Hecke eigenforms with the $g$ ideal classes of $R$, in the spirit of Shimura's 1972 construction of class fields over real quadratic fields from Hecke operators, whose numerical tables for prime levels $29\leq N\leq 233$ the paper reinterprets. However, the paper's Remark 3.4 observes that this class-field-theoretic behavior is special: for a general pseudo-Anosov foliation the extension $K|k$ need not be Galois, and even when it is, its Galois group need not be given by the ideal class group. The mechanism that forces $\mathrm{Gal}(K|k)\cong\mathrm{Cl}(R)$ in the modular-curve case is not identified, and no non-modular examples or counterexamples are given.

Problem Statement

Characterize, by explicit geometric or arithmetic conditions on a pseudo-Anosov measured foliation $\mathcal{F}$ on a compact surface, when the associated endomorphism-field extension $K=\mathbb{Q}\otimes\mathrm{End}(\mathrm{Jac}(\mathcal{F}))$ over the torus subfield $k=\mathbb{Q}\otimes\mathrm{End}(\mathrm{Jac}(h(\mathcal{F})))$ is Galois with Galois group isomorphic to the ideal class group $\mathrm{Cl}(R)$ of the order $R=\mathrm{End}(\mathrm{Jac}(h(\mathcal{F})))$ - that is, when the construction exhibits the ring-class-field behavior $\mathrm{Gal}(K|k)\cong\mathrm{Cl}(R)$ asserted by the source for the modular case. The only sufficient condition established in the source is that $\mathcal{F}$ be the foliation by vertical trajectories of a maximal Hecke eigenform on a modular curve $X_0(N)$. The objective decomposes into the following independently refutable sub-claims:

1. Characterization. Give explicit geometric or arithmetic conditions on $\mathcal{F}$ under which the extension $K|k$ is Galois with $\mathrm{Gal}(K|k)\cong\mathrm{Cl}(R)$.
2. Uniqueness of the modular case. Decide whether the modular-curve Hecke-eigenform case is essentially the only case, and whether the behavior occurs beyond it. This is implied by sub-claim 1, but may also be attacked directly.
3. Mechanism. Identify which features of the modular case are responsible for the class-field-theoretic behavior; this explains but does not by itself establish the characterization.
4. Examples. Produce explicit non-modular examples exhibiting or refuting the behavior (the source asserts non-universality without examples). A single example contributes to this sub-claim but does not by itself resolve sub-claim 1.
5. Conductor. Determine the missing conductor $\mathrm{N}(\mathfrak{c})$ (the norm of Shimura's ideal $\mathfrak{c}$) for the prime level $N=233$.

Sub-claim 5 is independent of sub-claims 1-4.

References

1. Igor V. Nikolaev, Measured Foliations and Hilbert 12th Problem, Acta Mathematica Vietnamica 48(2) (2023) 271-281, DOI 10.1007/s40306-022-00479-w, arXiv:0804.0057, https://arxiv.org/abs/0804.0057
2. Igor V. Nikolaev, Addendum to "Measured foliations and Hilbert 12th problem", arXiv:2505.22272 (2025), https://arxiv.org/abs/2505.22272
3. Goro Shimura, Class Fields Over Real Quadratic Fields and Hecke Operators, Annals of Mathematics 95(1) (1972) 130-190, DOI 10.2307/1970859, https://doi.org/10.2307/1970859
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