Classifying primitive Teichmüller curves in genera three and four

A Teichmüller curve is an algebraic curve in the moduli space $\mathcal M_g$ of genus-$g$ Riemann surfaces that is totally geodesic for the Teichmüller metric; it is generated by a Veech surface, a translation surface whose affine automorphism group is a lattice, and it is primitive when it does not arise from a covering construction on a lower-genus surface. McMullen's classification in genus two shows that every primitive Teichmüller curve in $\mathcal M_2$ lies in the union of the Weierstrass eigenform loci $W_D$ with a single exceptional curve, generated by the regular decagon, and that each $W_D$ is an explicitly understood finite union of Teichmüller curves. In the 2006 paper 'Prym varieties and Teichmüller curves' the same Prym-variety construction is pushed to genera three and four, producing infinitely many primitive Teichmüller curves in both, and the paper asks for the analogous classification: 'It would be interesting to have a similar analysis in genus 3 and 4.' Later work has settled large parts of this program: in genus three the minimal Prym stratum is completely classified (curves, components, cusps, orbifold points) and the two non-minimal Prym strata contain no primitive curves at all, while finiteness of the algebraically primitive curves is known, with effective-in-principle bounds far too large for a feasible enumeration; in genus four the Weierstrass loci $W_D(6)$ are single Teichmüller curves per discriminant with topological invariants determined, and the gothic curves form another infinite family. No classification of all primitive Teichmüller curves is known in either genus, and the genus-2-style statement — every primitive curve lies in a controlled union of loci up to an explicit finite exceptional list — remains open in both.

Background

A translation surface is a pair $(X,\omega)$ consisting of a compact Riemann surface $X$ and a nonzero holomorphic one-form $\omega$, regarded as a flat metric on $X$ with conical singularities at the zeros of $\omega$. Its group of affine automorphisms consists of homeomorphisms of $X$ that are affine in the flat coordinates and pull $\omega$ back to a scalar multiple of itself; when the derivatives of these automorphisms form a lattice in $\mathrm{SL}_2(\mathbb R)$, the surface is called a Veech surface and its $\mathrm{SL}_2(\mathbb R)$-orbit projects to an algebraic curve that is isometrically embedded for the Teichmüller metric — a Teichmüller curve $V\subset\mathcal M_g$. The trace field of $V$ is the number field generated by the traces of the derivatives of the affine automorphisms, equivalently by the ratios of the periods (path integrals) of $\omega$; its degree over $\mathbb Q$ is at most $g$. The curve is algebraically primitive when this degree equals $g$, and primitive when it is generated by a one-form of minimal genus, meaning it does not come from a covering construction built on a translation surface of lower genus. Teichmüller curves are the algebraic counterpart of rational billiards with optimal dynamics: a polygon whose angles are rational multiples of $\pi$ unfolds to a translation surface, and lattice polygons of genus $g$ yield primitive Teichmüller curves in $\mathcal M_g$.

In genus two the classification is complete. The Jacobian $\mathrm{Jac}(X)$ of a genus-two surface is an abelian surface, and it may carry real multiplication by the quadratic order $O_D$ of discriminant $D$ in the real quadratic field $\mathbb Q(\sqrt D)$, that is, an embedding of $O_D$ as a proper self-adjoint subring of the endomorphism ring of $\mathrm{Jac}(X)$; the form $\omega$ is an eigenform for this action. McMullen proved that every primitive Teichmüller curve in genus two arises from such real multiplication, classified the resulting eigenform loci by prototypes (combinatorial data recording the arithmetic of the real multiplication), and, using Möller's result on periodic points and Mordell-Weil ranks, showed that exactly one primitive Teichmüller curve lies outside the union of the Weierstrass loci $W_D$, the eigenform loci of forms with a single zero: the curve generated by the regular decagon, a form with two simple zeros whose Veech group is the $(2,5,\infty)$ triangle group. This is the 'analysis in genus two' to which the source question refers.

In genus two the Jacobian is itself the abelian surface carrying the real multiplication; in genera three and four the Jacobian has dimension three or four, and the role of the abelian surface carrying real multiplication is played instead by the Prym variety. For an involution $\rho:X\to X$ (a holomorphic self-map of order two other than the identity), the Prym variety $\mathrm{Prym}(X,\rho)$ is the complementary abelian subvariety, inside $\mathrm{Jac}(X)$, to the pullback of the Jacobian of the quotient curve $X/\rho$; its dimension is $g(X)-g(X/\rho)$. McMullen considers genus-three double covers of elliptic curves branched at four points and genus-four double covers of genus-two curves branched at two points: in both cases $\mathrm{Prym}(X,\rho)$ is an abelian surface that can carry real multiplication by $O_D$. The locus $\Omega E_D$ of Prym eigenforms — forms that are eigenvectors for this real multiplication — is $\mathrm{SL}_2(\mathbb R)$-invariant and nonempty only for $2\le g\le 5$; inside it, the Weierstrass locus $\Omega W_D$ consists of eigenforms whose zero is unique (of multiplicity $2g-2$) and is nonempty only for $g=2,3,4$. The main theorem of the source paper states that $W_D\subset\mathcal M_g$ is a finite union of Teichmüller curves, all primitive when $D$ is not a square, and that for $g=2,3,4$ there are infinitely many such curves. Their trace fields are the real quadratic field $\mathbb Q(\sqrt D)$, so in genera three and four these curves are primitive but not algebraically primitive; the algebraically primitive curves (trace field of degree $g$) form a separate, complementary part of the classification problem. Beyond the Prym construction, genus four also carries the gothic locus $\mathcal G$: the single $\mathrm{SL}_2(\mathbb R)$-invariant algebraic locus in the stratum $\Omega\mathcal M_4(2,2,2)$ of genus-four forms with three double zeros, discovered by McMullen, Mukamel and Wright, which is one locus rather than a discriminant-indexed family and whose projection to $\mathcal M_4$ is an infinite family of primitive Teichmüller curves disjoint from $\bigcup_D W_D$. The paper closes its classification discussion with the question that defines this problem: 'The primitive Teichmüller curves in genus 2 are classified in [Mc2] and [Mc3]; in particular, it is shown (using [Mo]) that there is only one such curve lying outside $\bigcup W_D\subset\mathcal M_2$. It would be interesting to have a similar analysis in genus 3 and 4.'

Problem Statement

Carry out for genera $g=3$ and $g=4$ the analysis of primitive Teichmüller curves that McMullen completed for $g=2$: a classification of all primitive Teichmüller curves in $\mathcal M_3$ and in $\mathcal M_4$ as lying in a controlled union of explicitly determined loci — that is, a union of $\mathrm{SL}_2(\mathbb R)$-invariant algebraic loci in $\mathcal M_g$ drawn from the following admissible class, each member of which is checkable by its defining condition: for each nonsquare discriminant $D$, the Prym eigenform locus $\Omega E_D$ (the locus of forms that are eigenvectors for real multiplication by $O_D$ on the Prym variety, whose single-zero locus is the Weierstrass locus $W_D$), together with the gothic locus $\mathcal G$ in genus four; a primitive Teichmüller curve belongs to the union exactly when it lies in one of these loci — up to an explicit finite list of further exceptional curves, each lying in none of the loci of the union, together with their arithmetic and topological invariants. Concretely: (1) for each nonsquare discriminant $D$, determine the Teichmüller curves comprising the Weierstrass loci $W_D\subset\mathcal M_3$ and $W_D\subset\mathcal M_4$, together with their arithmetic and topological invariants (trace fields, components, cusps, elliptic or orbifold points, Euler characteristics) — a part of the program that is now essentially complete, the curves in both loci and their invariants having been determined by Lanneau-Nguyen, Torres-Teigell-Zachhuber and Möller; and (2) decide whether every primitive Teichmüller curve in $\mathcal M_3$ (respectively $\mathcal M_4$) belongs to a controlled union of explicitly determined loci containing $\bigcup_D W_D$, up to an explicit finite list of further exceptional curves — the exact analogue of the genus-2 statement that only the curve generated by the regular decagon lies outside $\bigcup_D W_D\subset\mathcal M_2$, where in genus two the controlled union is $\bigcup_D W_D$ itself, while in genus four it must contain at least the gothic locus in addition, the gothic locus being an infinite primitive family disjoint from $\bigcup_D W_D$ — and identify which admissible loci carry primitive curves and which primitive curves form the exceptional list. Part (2) is the surviving open core: besides the covering statement, it requires the explicit determination, with their invariants, of the finitely many algebraically primitive Teichmüller curves in genus three, which lie outside the admissible loci and therefore appear on the exceptional list (finiteness is known by work of Bainbridge-Habegger-Möller, whose bounds are effective in principle but too large to yield an explicit list) and the placement of further known families such as the genus-four gothic curves among the curves outside $\bigcup_D W_D$. The intended completion boundary is the same as in genus two: a theorem listing all primitive Teichmüller curves of the genus, with enough data to pin down each one.

References

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