In two recent papers, E. Lanneau and D.-M. Nguyen, and M. Möller studied Teichmüller curves in genera and 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 ), 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 be a compact Riemann surface and let be a (non-trivial) holomorphic involution. The Prym variety is the Abelian variety

where (i.e., is the space of -anti-invariant holomorphic 1-forms on ) and (i.e., is the space of -anti-invariant cycles on ).

**Standing Hypothesis.** For the sake of today’s discussion, we will *always* assume that . 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 1Notice that , so

Remark 2By Riemann-Hurwitz formula, , so that our standing hypothesis and the previous remark imply . Moreover, when , one has that is unramified.

–**Real multiplication on Abelian varieties**–

Let be an integer such that or modulo , and let

be the real quadratic orderof discriminant .

A polarized Abelian variety can be thought as , where is a lattice equipped with a symplectic pairing . In this way, the endomorphism ring of can be thought as

We say that is *self-adjoint*if for all .

Definition 1We say that a polarized Abelian variety of complex dimension admits a real multiplication by if there exists a representation such that

- (a) is self-adjoint for all ;
- (b) is a proper subring of , i.e., if for some and , then .

–**Prym eigenforms**–

Let be a Prym variety admitting real multiplication by . We say that is a *Prym eigenform* whenever .

In the sequel, we denote by the locus of Prym eigenforms inside the moduli space of Abelian differentials on genus Riemann surfaces. Also, given a list of non-zero natural numbers such that , we denote by the locus of Prym eigenforms whose list of orders of its zeroes coincides with .

Remark 3In general, neither the holomorphic involution nor the representation are (uniquely) determined by the Prym eigenform . See, however, Theorem 5.1 of E. Lanneau and D.-M. Nguyen’s article for an uniqueness statement in genus .

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)is a closed -invariant locus of .

Following C. McMullen, we call *Weierstrass locus* the locus of Prym eigenforms inside the minimal stratum of (this name is motivated by the fact that its construction is naturally related to Weierstrass points).

Remark 4In the case of a Prym eigenform in the Weierstrass locus , its unique zero must be fixed by the holomorphic involution . On the other hand, by Remark 2, we know that and, moreover, doesn’t have fixed points when . Therefore, can occur only when .

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

Corollary 3 (McMullen)For , 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 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 , C. McMullen used these Teichmüller curves to *classify* the closures of the orbits of the natural action on the moduli space 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 . Here, is the Poincaré upper half-plane and is the lower half-plane (and the action of the subgroups of is through Moebius transformations).

A *special curve* or *Kobayashi geodesic* on is an algebraic curve that is totally geodesic with respect to the Kobayashi metric on .

A simple example of special curve on is the *diagonal* obtained by the composition of the map with the projection . In this article here, F. Hirzebruch and D. Zagier introduced the *twisted diagonals* obtained by the composition of the map with the projection , where and 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 . Roughly speaking, given , we can associated its Jacobian , where is the hyperelliptic involution of . By definition, whenever , one has that is a principally polarized Abelian variety with real multiplication by . Since the Hilbert modular surface parametrizes all principally polarized Abelian surfaces with real multiplication, and Teichmüller curves are Kobayashi geodesics, we can see naturally as a special curves on . Actually, C. McMullen showed that the embedding of on is given by the composition of a map of the form with the projection , where is holomorphic but it is *not* a Moebius transformation, i.e., is *not* given by some . In other words, the special curves induced by 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 , i.e., he considers the composition of the map with the projection , where and 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 or equivalently ).

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 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 , the Teichmüller curves forming the Weierstrass loci and 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 5Actually, the (genus ) case of can be treat without appealing to Lyapunov exponents: in fact, the Prym varieties arising from this case have a polarization of signature , so that they don’t lie in as it was defined in this post (as this last object parametrizes principally polarized Abelian surfaces). On the other hand, the (genus ) case of is non-trivial because the polarization has signature , i.e., it is a principal polarization, and hence the relevant Prym varieties lie on . In any event, we decided to include below an unified proof of the genus and 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 5The Teichmüller curves in 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 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 ), and Appendix D (for the case of ) 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 and 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 with a Model A+ cylinder decomposition and an Abelian differential in with a Model A cylinder decomposition (the other [analogous] cases left as an exercise to the reader).

Given a “Model A+” Abelian differential in , 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 , and . The subspace in absolute homology spanned by them has dimension (maximal) because is a canonical symplectic basis of homology (as one can see from the picture).

Similarly, given a “Model A” Abelian differential in , one can draw the (closed) cycles below

in order to see that the waist curves of horizontal cylinders (represented by and ) span a subspace of dimension (maximal) because is a canonical symplectic basis of homology.

Therefore, by G. Forni’s criterion, once we know that Teichmüller curves in and 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 ) case of Teichmüller curves inside , we know that the sum of the three non-negative Lyapunov exponents is

because is a subset of the connected component of Abelian differentials in 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 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 giving rise to the Prym variety over leads to a subbundle of the Hodge bundle (of dimension ) contributing with two explicit Lyapunov exponents: and . Therefore, given that the sum of the three exponents is , we conclude that the Lyapunov spectrum of Teichmüller curves in is

independently of the discriminant .

Remark 6As 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 whenever is a square, i.e., , by conveniently choosing parameters , and then using some programs by A. Zorich and V. Delecroix (available at SAGE after installation ofsage-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 squares) in with discriminant and choice of parameters , , , I got “numerical Lyapunov spectrum”:

In the (genus ) case of Teichmüller curves inside , we also know the sum of the four non-negative Lyapunov exponents is

because is a subset of the connected component of Abelian differentials in 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 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 of the Hodge bundle (of dimension ) giving rise to the Prym variety over contributes with two explicit Lyapunov exponents: and . In particular, the Lyapunov spectrum of Teichmüller curves in has the form

where .

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

Remark 7Analogously 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 whenever is a square by conveniently choosing parameters , and then using SAGE. For instance, the parameters , , , and lead to a square-tiled surface in with squares and “numerical Lyapunov spectrum”

and the parameters , , , , lead to a square-tiled surface in with squares and “numerical Lyapunov spectrum”

Personally, I tend to believe that these numerical experiments “indicate” that and , but, as I said, the precise values of and 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 where is a Teichmüller curve. Recall that, by definition, is a Prym eigenform, i.e., and for every . Since (by our standing hypothesis), we can take such that and for all (where is, as usual, the Galois conjugate of ). In this way, we can attach a number to by computing the orbifold degree of the line bundle over (and normalizing it by the Euler characteristic of ). For a characterization of in terms of intersection theory (of some divisors of with ), see Sections 1 and 5 of M. Möller’s paper. As it turns out, it is possible to show that is one of the Lyapunov exponents of the Kontsevich-Zorich cocycle over .

In the case of (genus ) Teichmüller curves in , this exponent is known to be always , while in the cases (of genus and resp.) of Teichmüller curves in and resp., M. Möller (see again Section 5 of his paper) showed that and resp.

At this point, the plan for the proof of Weiss’ theorem is simple: we will use the Lyapunov exponent as an “invariant” to distinguish twisted diagonals, twisted Teichmüller curves and , .

We begin by distinguishing Teichmüller curves in and from twisted Teichmüller curves. Since twisted Teichmüller curves are obtained by twisting Teichmüller curves (of genus ) in , it is not hard to check (i.e., the twisting operation doesn’t change this exponent). On the other hand, we just saw that Teichmüller curves in , resp. satisfy , resp. , so that the claim follows.

Now, we complete the sketch by distinguishing Teichmüller curves in and from twisted diagonals. As we mentioned above, the quantity admits an interpretation in terms of intersecion theory inside the Hilbert modular surface , and, as a matter of fact, the definition of can be naturally extended to any special curve (Kobayashi geodesic) in . Moreover, a direct computation reveals that this quantity (a sort of “slope”) equals 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 for twisted diagonals. Again, since for Teichmüller curves in and , we are done.

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