An algebraic description and classification of primitive Teichmüller curves in genus greater than two

A Teichmüller curve is the image in the moduli space $\mathcal{M}_g$ of Riemann surfaces of a closed orbit of the $GL_2^+(\mathbb{R})$ action on the bundle of holomorphic 1-forms; it is primitive when it does not arise from lower genus via a covering construction. McMullen's paper 'Teichmüller curves in genus two: Torsion divisors and ratios of sines' completed the classification of primitive Teichmüller curves in genus two and distilled it into an algebraic description: they are exactly the one-dimensional irreducible components of the loci $W_D[n]$ defined by real multiplication on the Jacobian together with a torsion condition on the divisor of the zeros. The paper's closing sentence asks for a similar algebraic description of the primitive Teichmüller curves in higher genus. Later work establishes the necessary algebraic constraints in all genera (real multiplication on a factor of the Jacobian, torsion divisors, algebraicity over number fields), but the eigenform locus for real multiplication on the full Jacobian fails to be $GL_2^+(\mathbb{R})$-invariant already in genus 3, and the known partial analogue uses real multiplication on a two-dimensional Prym factor: the Weierstrass curves in genus 3 and 4 (classified by Lanneau–Nguyen), sporadic examples, and the Bouw–Möller series. In genus 3 the Weierstrass curves account for all but finitely many primitive curves, by results of Eskin–Filip–Wright and Aulicino–Nguyen, but the exceptions are not enumerated and the completeness question for the three known sporadic curves (regular heptagon, regular 14-gon, and the $(2,3,4)$ triangle) is open; genus 4 has additional infinite families and no classification; in genus at least 5 only one construction is known and even infinitude is open.

Background

A translation surface is a pair $(X,\omega)$ where $X$ is a compact Riemann surface of genus $g$ and $\omega$ is a nonzero holomorphic 1-form on $X$; the form gives $X$ a flat metric with conical singularities at the zeros of $\omega$. Such pairs form a bundle $\Omega\mathcal{M}_g\to\mathcal{M}_g$ over the moduli space $\mathcal{M}_g$ of isomorphism classes of genus-$g$ Riemann surfaces, and the group $GL_2^+(\mathbb{R})$ acts on it by post-composing flat coordinates with linear maps. The projection of an orbit to $\mathcal{M}_g$ is usually dense. If the stabilizer $SL(X,\omega)\subset SL_2(\mathbb{R})$, called the Veech group, is a lattice (a discrete subgroup of finite covolume), then the quotient $V=\mathbb{H}/SL(X,\omega)$ of the upper half-plane $\mathbb{H}$ is a complex algebraic curve that maps isometrically into $\mathcal{M}_g$; its image is a Teichmüller curve, i.e. one of the closed $GL_2^+(\mathbb{R})$-orbits. By an unfolding construction, Teichmüller curves correspond to polygonal billiards with optimal dynamics (every trajectory periodic or uniformly distributed), which is one motivation for classifying them. A Teichmüller curve is primitive if it does not arise from a Teichmüller curve in some $\mathcal{M}_h$, $h<g$, via a covering construction. The trace field of a Teichmüller curve is the number field generated by traces of elements of its Veech group.

Genus two is the only genus where the primitive Teichmüller curves are completely classified. The classification, built up by Calta and McMullen in a series of papers and completed in the source paper (with the full orbit-closure classification in genus two also known, McMullen 2007), reads: every primitive Teichmüller curve in $\mathcal{M}_2$ is generated by a form with either a double zero — the countably many such curves are each determined by their discriminant and a spin invariant, a $\mathbb{Z}/2$-valued invariant from real multiplication — or by a form with simple zeros, in which case the curve is unique and generated by the regular-decagon form. The conceptual core of the completion is purely algebraic. Let $W_D[n]\subset\mathcal{M}_2$ be the locus of curves $X$ such that the Jacobian $\mathrm{Jac}(X)$ (the abelian variety dual to the space of holomorphic 1-forms modulo integral cycles) admits real multiplication by the order $\mathcal{O}_D$ of discriminant $D$ in a real quadratic field — a faithful action by symmetric endomorphisms — with the generating form an eigenform, i.e. an eigenvector for this action on the space of holomorphic 1-forms, and such that the divisor $[P-Q]$ of the two zeros of that eigenform has order dividing $n$ in $\mathrm{Jac}(X)$ (a torsion divisor condition). Then the primitive Teichmüller curves generated by genus-two forms are exactly the one-dimensional irreducible components of the loci $W_D[n]$, and those components can be effectively enumerated. Two facts drive the proof: Möller's theorem that along any Teichmüller curve whose trace field has degree equal to the genus — such curves are called algebraically primitive — differences of zeros of the generating form are torsion in the Jacobian; and the $GL_2^+(\mathbb{R})$-invariance in genus two of the eigenform locus for real multiplication, which forces one-dimensional components of $W_D[n]$ to be Teichmüller curves.

The source paper closes its introduction with: 'It would be interesting to have a similar algebraic description of the primitive Teichmüller curves in higher genus.' The obstacle is structural and known: for $g>2$ the eigenform locus for real multiplication on the full Jacobian is not $GL_2^+(\mathbb{R})$-invariant — already for $g=3$ this is McMullen's 'instability of real multiplication' theorem (2003, Theorem 7.5), so the genus-two argument cannot be transplanted verbatim. What is known in higher genus: Möller's torsion, real-multiplication (on a factor of the Jacobian), and algebraicity theorems hold in every genus as necessary conditions; the Prym construction — real multiplication on a two-dimensional Prym variety, the abelian subvariety of the Jacobian cut out by the $(-1)$-eigenspace of an involution of $X$ — yields the Weierstrass curves $W_D\subset\mathcal{M}_g$ for $g=3,4$, an infinite series of primitive Teichmüller curves; sporadic examples of $E_6$, $E_7$, $E_8$ type occur in genus 3 and 4, and in genus 3 the only three primitive curves known outside the Weierstrass series are generated by the regular heptagon, the regular 14-gon, and the $(2,3,4)$ triangle; the Bouw–Möller series $V_{p,q}$ provides finitely many examples in every $\mathcal{M}_g$ and is the only known construction in genus at least 5; and two further infinite families (gothic and arabesque) are known in genus 4. For genus 3 there is a near-completeness theorem: the Weierstrass curves account for all but finitely many primitive Teichmüller curves, a consequence of the orbit-closure structure theory of Eskin–Filip–Wright combined with Aulicino–Nguyen's classification of rank-two affine invariant submanifolds in genus 3 (an affine invariant submanifold is a closed $GL_2^+(\mathbb{R})$-invariant subvariety of a stratum, the strata partitioning $\Omega\mathcal{M}_g$ by the orders of the zeros of $\omega$; rank two refers to the dimension of the locally constant part of the natural cohomological cocycle over the action). But the finitely many exceptional genus-3 curves are not enumerated, no analogous statement exists in genus 4, and in genus at least 5 it is not even known whether infinitely many primitive Teichmüller curves exist.

Problem Statement

Give, for genus $g>2$, an algebraic description of the primitive Teichmüller curves analogous to the genus-two description, and use it to classify them. In the genus-two template (the source paper of this question, McMullen 2006, Theorems 1.6–1.7), the primitive Teichmüller curves are exactly the one-dimensional irreducible components of the loci $W_D[n]\subset\mathcal{M}_2$ defined by (i) real multiplication by the quadratic order $\mathcal{O}_D$ on the Jacobian, with the generating form an eigenform, and (ii) a torsion condition (order dividing $n$) on the divisor of the two zeros of the eigenform; and those components can be effectively enumerated. The higher-genus problem asks for an equally effective algebraic characterization — whether in terms of real multiplication on an abelian subvariety or factor (such as a Prym variety) together with torsion-type conditions on zeros, or through an equivalent structure — that identifies exactly the primitive Teichmüller curves in $\mathcal{M}_g$ for genus $g\ge 3$ (ultimately for every genus), together with the classification it yields: the complete list of such curves in genus $g\ge 3$, or a proof that the known families exhaust them up to explicitly bounded and enumerated exceptions. The concrete open instances recognized by the field are: (a) genus 3 — determine whether the Weierstrass series $W_D\subset\mathcal{M}_3$ together with the curves generated by the regular heptagon, the regular 14-gon, and the $(2,3,4)$ triangle accounts for all primitive Teichmüller curves, and make the known all-but-finitely-many statement effective; (b) genus 4 — classify the primitive Teichmüller curves in the presence of the gothic and arabesque families; (c) genus at least 5 — determine whether any primitive Teichmüller curves exist beyond the Bouw–Möller series, in particular whether infinitely many exist (Question 1.1 of McMullen's 2023 survey). These three instances are independently resolvable targets, and the general characterization for all $g>2$ is the umbrella goal that these instances realize in part.

References

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