Posted by: matheuscmss | January 12, 2012

Special curves in Hilbert modular surfaces and Lyapunov exponents of Prym eigenforms

In two recent papers, E. Lanneau and D.-M. Nguyen, and M. Möller studied Teichmüller curves in genera {3} and {4} steaming from Prym eigenforms. Their work can be seen as a sort of “follow-up” to C. McMullen’s seminal works (who completely treated these objects in genus {2}), and, from the point of view of Dynamical Systems, they are very interesting as a source of examples where the Lyapunov exponents of the Kontsevich-Zorich cocycle can be “described” (see, e.g., these links here for an introduction to the ergodic theory of the Kontsevich-Zorich cocycle). For instance, as it was noticed by M. Möller, C. Weiss and myself (independently), it is really easy to put together the works of D. Chen and M. Möller, A. Eskin, M. Kontsevich and A. Zorich, and M. Möller to compute the Lyapunov spectrum of these Teichmüller curves.

Of course, the knowledge of Lyapunov exponents per se may not seem very exciting, but during the Christmas Workshop 2011 of Karlsruhe University, C. Weiss gave a talk (about his PhD thesis under the supervision of M. Möller) showing how this information about Lyapunov exponents can be put forward to exhibit new special curves (i.e., Kobayashi geodesics) inside Hilbert modular surfaces. As it turns out, special curves in Hilbert modular surfaces are very rigid objects, and before C. Weiss’ result, the list of previously known special curves was not very long: it contained Hirzebruch-Zagier cycles (a.k.a. Shimura curves or twisted diagonals) and twisted Teichmüller curves solely.

The goal of today’s post is to revisit the construction of Teichmüller curves through Prym eigenforms and to give a (very rough) sketch of proof of C. Weiss’ theorem.

-Prym varieties-

Let {X} be a compact Riemann surface and let {\rho:X\rightarrow X} be a (non-trivial) holomorphic involution. The Prym variety {\textrm{Prym}(X,\rho)} is the Abelian variety

\displaystyle \textrm{Prym}(X,\rho):=(\Omega(X)^-)^*/H_1(X,\mathbb{Z})^-,

where {\Omega(X)^-:=\{\theta\in\Omega(X):\rho^*(\theta)=-\theta\}} (i.e., {\Omega(X)^-} is the space of {\rho}-anti-invariant holomorphic 1-forms on {X}) and {H_1(X,\mathbb{Z}):=\{\gamma\in H_1(X,\mathbb{Z}):\rho_*(\gamma)=-\gamma\}} (i.e., {H_1(X,\mathbb{Z})^-} is the space of {\rho}-anti-invariant cycles on {X}).

Standing Hypothesis. For the sake of today’s discussion, we will always assume that {\textrm{dim}_{\mathbb{C}}\textrm{Prym}(X,\rho)=2}. In fact, this assumption is “motivated” by C. McMullen’s work (see Section 3 of his article) on Prym varieties associated to genus 2 Riemann surfaces.

Remark 1 Notice that {\Omega(X/\rho)=\Omega(X)^+:=\textrm{Ker}(\rho^*-id)\subset\Omega(X)}, so

\displaystyle \textrm{dim}_{\mathbb{C}}\textrm{Prym}(X,\rho)=\textrm{dim}_{\mathbb{C}}\Omega(X)-\textrm{dim}_{\mathbb{C}}\Omega(X)^+ = g(X) - g(X/\rho)

Remark 2 By Riemann-Hurwitz formula, {g(X/\rho)\leq(g(X)+1)/2}, so that our standing hypothesis {\textrm{dim}_{\mathbb{C}}\textrm{Prym}(X,\rho)=2} and the previous remark imply {2\leq g(X)\leq 5}. Moreover, when {g(X)=5}, one has that {\rho} is unramified.

-Real multiplication on Abelian varieties-

Let {D>0} be an integer such that {D\equiv 0} or {1} modulo {4}, and let

\displaystyle \mathcal{O}_D\simeq \mathbb{Z}[X]/(X^2+bX+c)

be the real quadratic orderof discriminant {D=b^2-4c}.

A polarized Abelian variety {P} can be thought as {P\simeq\mathbb{C}^g/L}, where {L\simeq\mathbb{Z}^{2g}} is a lattice equipped with a symplectic pairing {\langle.,.\rangle:L\times L\rightarrow\mathbb{Z}}. In this way, the endomorphism ring {\textrm{End}(P)} of {P} can be thought as

\displaystyle \textrm{End}(P)\simeq\{T:\mathbb{C}^g\rightarrow\mathbb{C}^g: T(L)\subset L\}

We say that {T\in\textrm{End}(P)} is self-adjointif {\langle T(x),y\rangle = \langle x,T(y)\rangle} for all {x,y\in L}.

Definition 1 We say that a polarized Abelian variety {P} of complex dimension {\textrm{dim}_{\mathbb{C}}P=2} admits a real multiplication by {\mathcal{O}_D} if there exists a representation {i:\mathcal{O}_D\rightarrow\textrm{End}(P)} such that

  • (a) {i(\lambda)} is self-adjoint for all {\lambda\in\mathcal{O}_D};
  • (b) {i(\mathcal{O}_D)} is a proper subring of {\textrm{End}(P)}, i.e., if {nT\in i(\mathcal{O}_D)} for some {n\in\mathbb{N}-\{0\}} and {T\in\textrm{End}(P)}, then {T\in i(\mathcal{O}_D)}.

-Prym eigenforms-

Let {P=\textrm{Prym}(X,\rho)} be a Prym variety admitting real multiplication by {\mathcal{O}_D}. We say that {\omega\in\Omega(X)^-} is a Prym eigenform whenever {i(\mathcal{O}_D)\omega\subset\mathbb{C}\cdot\omega}.

In the sequel, we denote by {\Omega E_D} the locus of Prym eigenforms inside the moduli space {\Omega\mathcal{M}_g} of Abelian differentials on genus {g} Riemann surfaces. Also, given a list {(\kappa_1,\dots,k_{\sigma})} of non-zero natural numbers such that {\sum\limits_{j=1}^{\sigma}k_j=2g-2}, we denote by {\Omega E_D(k_1,\dots,\kappa_{\sigma})} the locus of Prym eigenforms whose list of orders of its zeroes coincides with {(k_1,\dots,k_{\sigma})}.

Remark 3 In general, neither the holomorphic involution {\rho} nor the representation {i:\mathcal{O}_D\rightarrow\textrm{End}(P)} are (uniquely) determined by the Prym eigenform {\omega}. See, however, Theorem 5.1 of E. Lanneau and D.-M. Nguyen’s article for an uniqueness statement in genus {3}.

The locus of Prym eigenforms is an important example in the dynamics of Teichmüller flow in view of the following theorem of C. McMullen:

Theorem 2 (McMullen) {\Omega E_D} is a closed {GL^+(2,\mathbb{R})}-invariant locus of {\Omega\mathcal{M}_g}.

Following C. McMullen, we call Weierstrass locus the locus {\Omega E_D(2g-2)} of Prym eigenforms inside the minimal stratum {\Omega\mathcal{M}_g(2g-2)} of {\Omega\mathcal{M}_g} (this name is motivated by the fact that its construction is naturally related to Weierstrass points).

Remark 4 In the case of a Prym eigenform in the Weierstrass locus {\Omega E_D(2g-2)}, its unique zero must be fixed by the holomorphic involution {\rho}. On the other hand, by Remark 2, we know that {2\leq g\leq 5} and, moreover, {\rho} doesn’t have fixed points when {g=5}. Therefore, {\Omega E_D(2g-2)\neq\emptyset} can occur only when {g=2,3,4}.

By combining McMullen’s theorem above with a “dimension counting” argument, it is possible to show that

Corollary 3 (McMullen) For {g=2,3,4}, {\Omega E_D(2g-2)} is a finite union of Teichmüller curves. Moreover, each of these Teichmüller curves are primitive (i.e., they are not obtained by branching covers of low genera Teichmüller curves) unless {D} is a square (i.e., unless the Teichmüller curve is associated to square-tiled surfaces, i.e., it is obtained by branched covers of the torus).

In the case of genus {2}, C. McMullen used these Teichmüller curves to classify the closures of the orbits of the natural {GL^+(2,\mathbb{R})} action on the moduli space {\Omega\mathcal{M}_2(2)} of Abelian differentials with a single double zero (a Teichmüller-theoretical analog to Ratner’s theorems). See this article of K. Calta, and these articles of C. McMullen for more details.

-Special curves on Hilbert modular surfaces-

Consider the Hilbert modular surface {X_D:=\mathbb{H}^2/SL(\mathcal{O}_D\oplus\mathcal{O}_D^{\vee})\simeq (\mathbb{H} \times\mathbb{H}^-)/SL_2(\mathcal{O}_D)}. Here, {\mathbb{H}} is the Poincaré upper half-plane and {\mathbb{H}^-} is the lower half-plane (and the action of the subgroups of {SL_2(\mathbb{R})} is through Moebius transformations).

A special curve or Kobayashi geodesic on {X_D} is an algebraic curve {C\rightarrow X_D} that is totally geodesic with respect to the Kobayashi metric on {X_D}.

A simple example of special curve on {X_D} is the diagonal obtained by the composition of the map {z\mapsto(z,z)} with the projection {\pi:\mathbb{H}^2\rightarrow X_D}. In this article here, F. Hirzebruch and D. Zagier introduced the twisted diagonals obtained by the composition of the map {z\mapsto (Mz,M^{\sigma}z)} with the projection {\pi:\mathbb{H}^2\rightarrow X_D}, where {M\in GL_2(\mathbb{Q}(\sqrt{D}))} and {M^{\sigma}} is its Galois conjugate. In the literature, twisted diagonals are known as Hirzebruch-Zagier cycles or Shimura curves.

More examples of special curves are given by the Teichmüller curves in genus 2 provided by McMullen’s Weierstrass loci {\Omega E_D(2)}. Roughly speaking, given {(X,\omega)\in\Omega E_D(2)}, we can associated its Jacobian {Jac(X)\simeq\textrm{Prym}(X,\rho)}, where {\rho} is the hyperelliptic involution of {X}. By definition, whenever {(X,\omega)\in\Omega E_D(2)}, one has that {\textrm{Prym}(X,\rho)} is a principally polarized Abelian variety with real multiplication by {\mathcal{O}_D}. Since the Hilbert modular surface parametrizes all principally polarized Abelian surfaces with real multiplication, and Teichmüller curves are Kobayashi geodesics, we can see {\Omega E_D(2)} naturally as a special curves on {X_D}. Actually, C. McMullen showed that the embedding of {\Omega E_D(2)} on {X_D} is given by the composition of a map of the form {z\mapsto (z,\phi(z))} with the projection {\pi:\mathbb{H}^2\rightarrow X_D}, where {\phi} is holomorphic but it is not a Moebius transformation, i.e., {\phi} is not given by some {M\in GL_2(\mathbb{Q}(\sqrt{D}))}. In other words, the special curves induced by {\Omega E_D(2)} are not twisted diagonals. By following F. Hirzebruch and D. Zagier, C. Weiss considers in his PhD thesis (under the supervision of M. Möller) twisted versions of these Teichmüller curves on {X_D}, i.e., he considers the composition of the map {z\mapsto (Mz,M^{\sigma}\phi(z))} with the projection {\pi:\mathbb{H}^2\rightarrow X_D}, where {M\in GL_2^+(\mathbb{Q}(\sqrt{D}))} and {\phi} is the holomorphic non-Moebius map described above. In this setting, C. Weiss showed that these twisted Teichmüller curves are special curves with finite area and he provides information on their stabilizer (inside {SL(\mathcal{O}_D\oplus\mathcal{O}^{\vee}_D)} or equivalently {SL_2(\mathcal{O}_D)}).

In any event, the fact that special curves are totally geodesic with respect to Kobayashi metric indicates that they are very rigid objects. In particular, one could ask whether all special curves inside {X_D} are twisted diagonals or twisted Teichmüller curves. The answer to this question is provided by the following result of C. Weiss:

Theorem 4 (C. Weiss) For every {D}, the Teichmüller curves forming the Weierstrass loci {\Omega E_D(4)} and {\Omega E_D(6)} are neither twisted diagonals nor twisted Teichmüller curves.

The proof of this result depends on the precise knowledge of some Lyapunov exponents of the Kontsevich-Zorich cocycle over these Teichmüller curves. So, before entering into a sketch of proof of this theorem, let us quickly discuss Lyapunov exponents of Kontsevich-Zorich (KZ) cocycle of Teichmüller curves on the Weierstrass locus (for a brief review of Lyapunov exponents of KZ cocycle, please see these posts here).

Remark 5 Actually, the (genus {3}) case of {\Omega E_D(4)} can be treat without appealing to Lyapunov exponents: in fact, the Prym varieties arising from this case have a polarization of signature {(2,1)}, so that they don’t lie in {X_D} as it was defined in this post (as this last object parametrizes principally polarized Abelian surfaces). On the other hand, the (genus {4}) case of {\Omega E_D(6)} is non-trivial because the polarization has signature {(2,2)}, i.e., it is a principal polarization, and hence the relevant Prym varieties lie on {X_D}. In any event, we decided to include below an unified proof of the genus {3} and {4} cases just to show the role of played by Lyapunov exponents.

-Lyapunov spectrum of Teichmüller curves in the Weierstrass locus-

We begin with the following result of E. Lanneau and D.-M. Nguyen:

Theorem 5 The Teichmüller curves in {\Omega E_D(4)} always contain an Abelian differential whose (periodic) horizontal foliation has a decomposition into (maximal) cylinders accordingly to Model A+, Model A- or Model B below

The Teichmüller curves in {\Omega E_D(6)} always contain an Abelian differential wihose (periodic) horizontal foliation has a decomposition into (maximal) cylinders accordingly to Model A or Model B below

For a proof (and the original pictures!) of this theorem, see Proposition 3.2, Corollary 3.4 (for the case of {\Omega E_D(4)}), and Appendix D (for the case of {\Omega E_D(6)}) of E. Lanneau and D.-M. Nguyen’s article.

A consequence of this theorem is the fact that, in the language of this article of G. Forni, Teichmüller curves in the Weierstrass loci {\Omega E_D(4)} and {\Omega E_D(6)} have maximal homological dimension, i.e., these Teichmüller curves contain some Abelian differential whose (periodic) horizontal foliation has a decomposition into (maximal) cylinders so that their waist curves generates a Lagrangian subspace (with respect to the symplectic intersection form) in the absolute homology group. Indeed, let us check that the homological dimension is maximal in the cases of an Abelian differential in {\Omega E_D(4)} with a Model A+ cylinder decomposition and an Abelian differential in {\Omega E_D(6)} with a Model A cylinder decomposition (the other [analogous] cases left as an exercise to the reader).

Given a “Model A+” Abelian differential in {\Omega E_D(4)}, one can follow E. Lanneau and D.-M. Nguyen and draw the (closed) cycles below:

In this picture, the waist curves of horizontal cylinders are represented by the cycles {\alpha_{1,1}}, {\alpha_1} and {\alpha_{2,1}}. The subspace in absolute homology spanned by them has dimension {3} (maximal) because {\{\alpha_{1,1},\beta_{1,1},\alpha_1,\beta_1,\alpha_{2,1},\beta_{2,1}\}} is a canonical symplectic basis of homology (as one can see from the picture).

Similarly, given a “Model A” Abelian differential in {\Omega E_D(6)}, one can draw the (closed) cycles below

in order to see that the waist curves of horizontal cylinders (represented by {\alpha_{1,1},\alpha_{2,1},\alpha_{2,2}} and {\alpha_{1,2}}) span a subspace of dimension {4} (maximal) because {\{\alpha_{1,1},\beta_{1,1},\alpha_{2,1},\beta_{2,1},\alpha_{2,2},\beta_{2,2},\alpha_{1,2},\beta_{1,2}\}} is a canonical symplectic basis of homology.

Therefore, by G. Forni’s criterion, once we know that Teichmüller curves in {\Omega E_D(4)} and {\Omega E_D(6)} have maximal homological dimension, we have that the KZ cocycle over this Teichmüller curves is non-uniformly hyperbolic, i.e., none of the Lyapunov exponents of KZ cocycle over them is zero.

Actually, as it was mentioned in the introduction to this post, one can say more about these Lyapunov exponents due to the recent works of D. Chen and M. Möller, A. Eskin, M. Kontsevich and A. Zorich, and M. Möller.

In the (genus {3}) case of Teichmüller curves inside {\Omega E_D(4)}, we know that the sum of the three non-negative Lyapunov exponents {1=\lambda_1>\lambda_2\geq\lambda_3(\geq 0)} is

\displaystyle 1+\lambda_2+\lambda_3 = 8/5

because {\Omega E_D(4)} is a subset of the connected component {\Omega\mathcal{M}_3(4)^{\textrm{odd}}} of Abelian differentials in {\Omega\mathcal{M}_3(4)} with odd spin structure (see Remark 1.2 of E. Lanneau and D.-M. Nguyen’s article or Lemma 2.1 of M. Möller’s article), and the sum of Lyapunov exponents of Teichmüller curves in this connected component is always {8/5} as it was shown by D. Chen and M. Möller(with the aid of a beautiful formula derived by A. Eskin, M. Kontsevich and A. Zorich relating sum of exponents with slopes of divisors in moduli spaces). Moreover, M. Möller (see Section 5 of his recent paper) was able to show that the subbundle {\Omega(X)^-} giving rise to the Prym variety {\textrm{Prym}(X,\rho)} over {\Omega E_D(4)} leads to a subbundle of the Hodge bundle (of dimension {2}) contributing with two explicit Lyapunov exponents: {1} and {1/5}. Therefore, given that the sum of the three exponents is {8/5}, we conclude that the Lyapunov spectrum of Teichmüller curves in {\Omega E_D(4)} is

\displaystyle \{1,2/5,1/5\}

independently of the discriminant {D}.

Remark 6 As a “side remark”, let me point out that one can “numerically test” these exponents by using (say) Proposition 4.2 of E. Lanneau and D.-M. Nguyen paper to build up explicit square-tiled surfaces (with “Model A+” cylinder decomposition) in {\Omega E_D(4)} whenever {D} is a square, i.e., {\sqrt{D}\in\mathbb{N}}, by conveniently choosing parameters {(w,h,t,e)\in\mathbb{Z}^4}, and then using some programs by A. Zorich and V. Delecroix (available at SAGE after installation of sage-combinatpackage). I plan to make some comments on this program later (in future posts), but for now let me say that by testing a square-tiled surface (with {28} squares) in {\Omega E_D(4)} with discriminant {D=49} and choice of parameters {w=6}, {h=1}, {t=0}, {e=1} I got “numerical Lyapunov spectrum”:

\displaystyle \{1., 0.39926717019771224, 0.2012323208392183\}

In the (genus {4}) case of Teichmüller curves inside {\Omega E_D(6)}, we also know the sum of the four non-negative Lyapunov exponents is

\displaystyle 1+\mu_2+\mu_3+\mu_4 = 2 = 14/7

because {\Omega E_D(6)} is a subset of the connected component {\Omega\mathcal{M}_4(6)^{\textrm{even}}} of Abelian differentials in {\Omega\mathcal{M}_4(6)} with even spin structure (see Proposition 2.2 of M. Möller’s paper), and the sum of Lyapunov exponents of Teichmüller curves in this connected component is always {14/7} as it was shown by D. Chen and M. Möller. Moreover, M. Möller (see again Section 5 of his recent paper) proved that the subbundle {\Omega(X)^-} of the Hodge bundle (of dimension {2}) giving rise to the Prym variety {\textrm{Prym}(X,\rho)} over {\Omega E_D(6)} contributes with two explicit Lyapunov exponents: {1} and {1/7}. In particular, the Lyapunov spectrum of Teichmüller curves in {\Omega E_D(6)} has the form

\displaystyle \{1,\alpha,\beta,1/7\}

where {\alpha+\beta=6/7}.

Here, I don’t know how to determine the precise values of {\alpha} and {\beta}, but this will not be important for the sequel.

Remark 7 Analogously to Remark 6 above, one can “numerically test” these exponents by using (say) Proposition D.2 of E. Lanneau and D.-M. Nguyen paperto build up explicit square-tiled surfaces (with “Model A” cylinder decomposition) in {\Omega E_D(6)} whenever {D} is a square by conveniently choosing parameters {(w,h,t,e)\in\mathbb{Z}^4}, and then using SAGE. For instance, the parameters {w=12}, {h=1}, {t=0}, {e=1} and {D=49} lead to a square-tiled surface in {\Omega E_{49}(6)} with {56} squares and “numerical Lyapunov spectrum”

\displaystyle \{1., 0.58989609784948371, 0.26631716403855049, 0.14294287190482519\}

and the parameters {w=20}, {h=1}, {t=0}, {e=1}, {D=81} lead to a square-tiled surface in {\Omega E_{81}(6)} with {90} squares and “numerical Lyapunov spectrum”

\displaystyle \{1.,0.59421161410520018, 0.26704624488102718, 0.14468572140958874\}

Personally, I tend to believe that these numerical experiments “indicate” that {\alpha=4/7} and {\beta=2/7}, but, as I said, the precise values of {\alpha} and {\beta} are unknown.

In any case, we are ready to say a few words on the proof of C. Weiss’ theorem 4.

-Quick sketch of proof of Weiss’ theorem-

Let {(X,\omega)\in C\subset \Omega E_D} where {C} is a Teichmüller curve. Recall that, by definition, {\omega} is a Prym eigenform, i.e., {\omega\in\Omega(X)^-} and {i(\lambda)\omega=\lambda\omega} for every {\lambda\in\mathcal{O}_D}. Since {\textrm{Prym}(X,\rho)=2} (by our standing hypothesis), we can take {\omega^{\sigma}\in\Omega(X)^-} such that {\Omega(X)^-=\mathbb{C}\omega\oplus\mathbb{C}\omega^{\sigma}} and {i(\lambda)\omega^{\sigma}=\lambda^{\sigma}\omega^{\sigma}} for all {\lambda\in\mathcal{O}_D} (where {\lambda^{\sigma}} is, as usual, the Galois conjugate of {\lambda}). In this way, we can attach a number {\lambda(C)} to {C} by computing the orbifold degree of the line bundle {\mathbb{C}\omega^{\sigma}} over {C} (and normalizing it by the Euler characteristic of {C}). For a characterization of {\lambda(C)} in terms of intersection theory (of some divisors of {X_D} with {C}), see Sections 1 and 5 of M. Möller’s paper. As it turns out, it is possible to show that {0\leq \lambda(C)<1} is one of the Lyapunov exponents of the Kontsevich-Zorich cocycle over {C}.

In the case of (genus {2}) Teichmüller curves in {\Omega E_D(2)}, this exponent is known to be always {1/3}, while in the cases (of genus {3} and {4} resp.) of Teichmüller curves in {\Omega E_D(4)} and {\Omega E_D(6)} resp., M. Möller (see again Section 5 of his paper) showed that {\lambda(C)=1/5} and {\lambda(C)=1/7} resp.

At this point, the plan for the proof of Weiss’ theorem is simple: we will use the Lyapunov exponent {\lambda(C)} as an “invariant” to distinguish twisted diagonals, twisted Teichmüller curves and {\Omega E_D(4)}, {\Omega E_D(6)}.

We begin by distinguishing Teichmüller curves in {\Omega E_D(4)} and {\Omega E_D(6)} from twisted Teichmüller curves. Since twisted Teichmüller curves {C} are obtained by twisting Teichmüller curves (of genus {2}) in {\Omega E_D(2)}, it is not hard to check {\lambda(C)=1/3} (i.e., the twisting operation doesn’t change this exponent). On the other hand, we just saw that Teichmüller curves {C} in {\Omega E_D(4)}, resp. {\Omega E_D(6)} satisfy {\lambda(C)=1/5}, resp. {\lambda(C)=1/7}, so that the claim follows.

Now, we complete the sketch by distinguishing Teichmüller curves in {\Omega E_D(4)} and {\Omega E_D(6)} from twisted diagonals. As we mentioned above, the quantity {\lambda(C)} admits an interpretation in terms of intersecion theory inside the Hilbert modular surface {X_D}, and, as a matter of fact, the definition of {\lambda(C)} can be naturally extended to any special curve (Kobayashi geodesic) in {X_D}. Moreover, a direct computation reveals that this quantity (a sort of “slope”) equals {1} for the diagonal, and, since it can be checked that the twisting operation doesn’t change this quantity, one get that this quantity is also {1} for twisted diagonals. Again, since {\lambda(C)<1} for Teichmüller curves in {\Omega E_D(4)} and {\Omega E_D(6)}, we are done.


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