Explicit models for isoperiodic moduli spaces: the boundary of the periodic locus and genus-three cubic-field examples

The moduli space $A(L)$ collects the genus-$g$ translation surfaces whose absolute periods form a fixed polarized module $L$; it is a union of leaves of the absolute period foliation of the Hodge bundle. The source paper showed that $A(L)$ is a complete stratified Euclidean slice of the Schottky locus in Siegel space, and developed its genus-two geometry in detail, with an essentially complete picture for the eigenform lattice of discriminant $D=5$: an explicit two-parameter enumeration of the zeros and poles of the associated quadratic differential, a classification of fake pentagons, a bijection between pole orbits and strict ideal classes, and an explicit family whose bifurcation set is homeomorphic to the ordinal $\omega^\omega$. The paper's flagship question - connectedness of $A(L)$ for genus $g \ge 4$ - has since been resolved affirmatively by Calsamiglia, Deroin and Francaviglia via a transfer principle for the period map; the companion ergodicity question is settled, and the genus-two ordinal bound $\omega^\omega$ is proven and sharp. These resolved parts are not part of this task. The surviving open core consists of the two explicit descriptions the paper itself asked for and left unfinished: (1) an explicit topological model for the boundary $\partial A_0(L)$ of the periodic locus, a countable union of half-infinite leaves known to be non-locally-connected for eigenform lattices; and (2) a detailed investigation of the genus-three isoperiodic spaces, isomorphic to the degree-two Siegel space, carrying the natural action of $SL_2(\mathcal{O}_K \oplus \mathcal{O}_K^\vee)$ for totally real cubic fields $K$.

Background

A translation surface is a compact Riemann surface $X$ together with a nonzero holomorphic one-form $\omega$: integrating $\omega$ gives charts in which $X$ is flat, with cone singularities at the zeros of $\omega$. The moduli space $\Omega\mathcal{M}_g$ of such pairs is called the Hodge bundle; postcomposing the flat charts with linear maps gives the $SL_2(\mathbb{R})$-action of Teichmüller dynamics. The absolute periods of $(X,\omega)$ are the integrals $\mathrm{Per}(X,\omega) = \{\int_\gamma \omega : \gamma \in H_1(X,\mathbb{Z})\} \subset \mathbb{C}$, a free abelian group of rank at most $2g$ carrying the symplectic intersection pairing; a free rank-$2g$ submodule $L \subset \mathbb{C}$ with a unimodular symplectic form is called a polarized module. The absolute period foliation (or isoperiodic foliation) of $\Omega\mathcal{M}_g$ consists of the deformations of $(X,\omega)$ that fix every absolute period; its leaves have complex dimension $2g-3$. Fixing the isomorphism class of the period module rather than the periods pointwise gives the moduli space $A(L) = \{(X,\omega)$ stable : $\mathrm{Per}(X,\omega) \cong L$ as polarized modules$\}$, taken inside the compactification by stable forms (forms on nodal curves, nonzero on every component).

McMullen (2014) proved that $A(L)$ is isomorphic, as a complex variety, to a standard slice $S^g \cap \mathbb{H}_{g-1}$ of the Schottky locus in Siegel space $\mathbb{H}_{g-1}$ (the symmetric complex $(g-1)\times(g-1)$ matrices with positive definite imaginary part, parametrizing principally polarized abelian varieties), where $S^g$ is the locus of period matrices of genus-$g$ curves, whose characterization is the classical Schottky problem. Relative period coordinates give $A(L)$ a stratified complex Euclidean structure with a complete path metric, and $A(L) \cong \mathbb{H}_{g-1}$ for $g \le 3$. For $g \ge 4$ the slice topology is nontrivial, which is why the paper's flagship question was whether $A(L)$ is connected; this has since been resolved affirmatively.

Genus two. Here $A(L) \cong \mathbb{H}$ (the upper half-plane) carries a natural meromorphic quadratic differential $q(L) = dt_1^2$ whose flat coordinates record the Euclidean structure: its simple zeros are the points of $W(L)$, the forms with a double zero, and its poles are the points of $P(L)$, the singular curves. Level sets of $\mathrm{Im}(t_1)$ define a foliation $F(L)$ of $A(L)$: nearby forms on one leaf are related by horizontally sliding the zeros while keeping all absolute periods fixed. The periodic locus $A_0(L)$ consists of the forms whose horizontal foliation is periodic (all leaves closed, so the form decomposes into flat cylinders); it is nontrivial, $\emptyset \ne A_0(L) \ne A(L)$, exactly when $L \cap \mathbb{R}$ is a Lagrangian submodule of $L$ (rank $g$, isotropic for the symplectic form), the zero flux condition. In that case $A_0(L)$ is an open dense topological disk tiled by rectangular strips $R_i$ of heights $h_i$; the heights are drawn from the spectrum $\mathrm{Spec}(L)$, the discrete set of sums $y_1+y_2$ over positive bases $(y_1,y_2)$ of $\mathrm{Im}(L)$, computable by the Euclidean (continued fraction) algorithm. The boundary $\partial A_0(L)$ is the union $\bigcup_{f>0} A_f(L)$ of countably many half-infinite leaves, indexed by the upper degree $f = \deg^+(X,\omega)$, an integer invariant of the topological type of the foliation with $f = 0$ exactly on the periodic locus, characterized by $f = [\mathrm{Im}(L):\mathrm{Im}(S)]$ when $(X,\omega)$ is a connected sum of two genus-one forms with period submodules $S$ and $S^\perp$; the upper degree strictly increases under limits (equation (4.3) of the source). For eigenform lattices $L_D$ the boundary is topologically wild: boundary points of sufficiently large upper degree are inaccessible, so $\partial A_0(L_D)$ is not locally connected (Theorem 1.9), and the paper states that it would be interesting to find an explicit model for $\partial A_0(L)$ as a topological space.

Eigenforms and Hilbert modular surfaces. A genus-two form $(X,\omega)$ is an eigenform for real multiplication by the quadratic order $\mathcal{O}_D = \mathbb{Z}[(D+\sqrt{D})/2]$ of discriminant $D > 0$ if its Jacobian admits an embedding $\mathcal{O}_D \to \mathrm{End}(\mathrm{Jac}(X))$ under which $[\omega]$ is an eigenvector; equivalently, up to the action of $GL_2^+(\mathbb{R})$, the period module is the eigenform lattice $L_D = \mathcal{O}_D \oplus \mathcal{O}_D i$ of equation (1.3) of the source, and this is exactly the case in which the $SL_2(\mathbb{R})$-saturation of $A(L)$ is closed. The geometry is then governed by the Hilbert modular surface $X_D = (\mathbb{H}\times\mathbb{H})/SL_2(\mathcal{O}_D \oplus \mathcal{O}_D^\vee)$, with $\mathcal{O}_D^\vee$ the trace dual module of $\mathcal{O}_D$, the quotient parametrizing abelian surfaces with real multiplication by $\mathcal{O}_D$, on which the images of $W(L_D)$ and $P(L_D)$ are curves $W_D$ and $P_D$. The foliation $F(L_D)$ is preserved by the affine group $AN_2(\mathcal{O}_D)$ of upper triangular matrices in $SL_2(\mathcal{O}_D)$, that is the matrices $\pm\begin{pmatrix}\epsilon^{-n} & b \\ 0 & \epsilon^n\end{pmatrix}$ with $n \in \mathbb{Z}$, $b \in \mathcal{O}_D$ and $\epsilon$ the fundamental unit; this discrete group substitutes for $SL_2(\mathbb{R})$, and the orbit spaces $W(L_D)/AN_2(\mathcal{O}_D)$ and $P_f(L_D)/AN_2(\mathcal{O}_D)$, where $P_f(L) = P(L) \cap A_f(L)$ is the set of poles of upper degree $f$, are finite for each $f \ge 0$ (Theorem 1.6).

The case $D = 5$. Writing $\epsilon = (1+\sqrt{5})/2$, one has $\mathrm{Spec}(L_D) = \epsilon^{\mathbb{Z}}$ and a single splitting type. Apart from a distinguished zero $W_0$ with coordinate $\tau_2(W_0) = i\sqrt{D}$ representing an eigenform for complex multiplication, the zeros of $q(L_D)$ form a single $AN_2(\mathcal{O}_D)$-orbit $\{W_{n,k}\}$ with $Y(W_{n,k}) = \epsilon^n$ for the height function $Y : A_0(L) \to (0,\infty)$, and each zero $W_{n,k}$ is joined by a leaf to a unique pole $P_{n,k}$ with $\tau_2(P_{n,k}) = \epsilon^{4n-2}\tau_1 + 5k\epsilon^{2n-1}$, $\tau_1 = i\sqrt{5}$ (equation (1.4)). A fake pentagon is a genus-two form with a double zero and period module $\mathbb{Z}[\zeta_5]$, the regular pentagon form being one of them; every fake pentagon is uniquely of the form $A \cdot W_{k,n}$ (Theorem 9.1). The pole orbits are counted by strict class numbers: for $f > 0$ there is a natural bijection $P_f(L_D)/AN_2(\mathcal{O}_D) \cong \mathrm{Pic}^+(\mathcal{O}_{f^2D})$ with the strict (oriented) ideal class group of the order $\mathcal{O}_{f^2D} = \mathbb{Z} + f\,\mathcal{O}_D$ of conductor $f$, so the number of orbits is the strict class number $h^+(f^2D)$ (Theorem 10.1). Finally, an explicit genus-two family $(X_t,\omega_t)$ has bifurcation set $B = \{t : F(i\omega_t)$ is not periodic$\}$ homeomorphic to the ordinal $\omega^\omega$ (Theorem 11.1); here derived sets (iterated boundaries, $B^{(k+1)} = \partial B^{(k)}$) are taken transfinitely, and $\omega^\omega$ is the first ordinal of infinite Cantor-Bendixson rank, the canonical space obtained from a countable closed set by iterating the derivative infinitely often. The paper notes $B$ can be no more complex than this ordinal, by equation (4.3).

Genus three. For $g = 3$ one has $A(L) \cong \mathbb{H}_2$, the degree-two Siegel space, and for any totally real cubic field $K$ (degree three over $\mathbb{Q}$, with three real embeddings) there is a natural action of $SL_2(\mathcal{O}_K \oplus \mathcal{O}_K^\vee)$ on $\mathbb{H}_2$ (Proposition 1.11). In contrast to the quadratic case, most points of $\mathbb{H}_2$ do not represent eigenforms, and the paper asks for these genus-three examples to be investigated in more detail.

Later developments. Calsamiglia, Deroin and Francaviglia introduced a transfer principle relating isoperiodic leaves to the fibers of the period map $\mathrm{Per}_g$ on the Torelli cover of moduli space (homologically marked curves), where the primitive degree of a period point is the degree of the induced map to its image as a polarized abelian variety, infinite when the image is non-discrete; Hamenstädt, Winsor, Ygouf and others developed the ergodic theory of these foliations at the level of strata and of meromorphic forms. The macroscopic questions (connectedness, ergodicity, closure classification, ordinal bounds in genus two) are settled; what remains open is the explicit local theory stated below.

Problem Statement

Let $L \subset \mathbb{C}$ be a polarized module of rank $2g$ and let $A(L)$, $A_0(L)$, $F(L)$, $q(L)$, $W(L)$, $P_f(L)$, the upper degree $\deg^+$, the spectrum $\mathrm{Spec}(L)$ and the height function $Y$ be as defined in McMullen (2014) and recalled in the background. The explicit questions that paper posed and left open, asking for the explicit description of isoperiodic moduli spaces beyond what was established for discriminant $D = 5$ in genus two, comprise two objectives.

(P1) Boundary topology of the periodic locus. For a zero-flux module $L$ (equivalently, $\emptyset \ne A_0(L) \ne A(L)$), give an explicit model of $\partial A_0(L)$ as a topological space: a concretely described space, together with a proof that it is homeomorphic to $\partial A_0(L)$, consistent with (i) the decomposition $\partial A_0(L) = \bigcup_{f>0} A_f(L)$ into countably many half-infinite leaves each containing a unique pole of $q(L)$, (ii) the strict increase of the upper degree under limits (equation (4.3) of the source), and (iii) Theorem 1.9, the inaccessibility of boundary points of large upper degree and the non-local-connectivity of $\partial A_0(L_D)$.

(P2) Genus-three cubic examples. For a totally real cubic field $K$ with ring of integers $\mathcal{O}_K$: investigate in more detail the isoperiodic moduli spaces $A(L) \cong \mathbb{H}_2$ carrying the natural $SL_2(\mathcal{O}_K \oplus \mathcal{O}_K^\vee)$-action of Proposition 1.11. As the source notes, in contrast to the quadratic case most points of $\mathbb{H}_2$ do not represent eigenforms; what is unknown here includes characterizing the eigenform locus inside $\mathbb{H}_2$, describing the analogues of the zero and pole loci and their orbit structure under the arithmetic group, and determining the topology of the periodic locus and of its boundary in this setting.

Both objectives reproduce questions explicitly posed in the source: the boundary-model request of the Introduction, immediately after the discussion of Theorem 1.4 ('It would be interesting to find an explicit model for $\partial A_0(L)$ as a topological space'), with Theorem 1.9 establishing the non-local-connectivity of $\partial A_0(L_D)$ that such a model must reflect; and the genus-three invitation following Proposition 1.11 ('It would be interesting to investigate these genus three examples in more detail'). The paper does not pose as open questions the extension of the $D = 5$ enumeration (equation (1.4)), of the fake-pentagon classification (Theorem 9.1) or of the class-number correspondence (Theorem 10.1) to other discriminants, nor a closed-form question for the strip heights; those natural directions are related background rather than objectives or completion requirements.

References

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