# Torsion on Jacobians and Teichmuller dynamics

In this post, I will describe a special case of a theorem about torsion points on Jacobians and Teichmuller dynamics which is proved in this paper. Namely, I will describe a different proof of a theorem of Moeller that for Teichmuller curves, zeroes of the holomorphic ${1}$-form must land in the torsion of (a factor of) the Jacobian. Some familiarity with Teichmuller dynamics will be assumed, as in the survey of Zorich.

— 1. Setup —

Consider a Teichmuller curve ${X=\Gamma\setminus {\mathbb H}}$, coming from a closed orbit of the ${{\mathrm{SL}}_2{\mathbb R}}$-action on a stratum of flat surfaces ${{\mathcal H}}$. We have a family of Riemann surfaces over ${X}$ denoted ${C{\rightarrow} X}$ and denote by ${C_x}$ the fiber corresponding to a point ${x\in X}$. We also have choice of holomorphic ${1}$-form, i.e. a line ${\langle \omega_x\rangle\subseteq H^{1,0}(C_x)}$. Note that on ${X}$ we only have the ${1}$-forms ${\omega}$ up to scaling, whereas on the tangent space ${TX}$ we actually have a choice of ${1}$-form.

The real and imaginary parts of ${\omega_x}$ give a local system ${H^1_{\iota_0}}$ over ${X}$. By work of Moeller, this is defined over a totally real number field ${k}$. For any other embedding ${\iota:k{\rightarrow} {\mathbb R}}$ we have a corresponding local system ${H^1_\iota}$, which is a Galois-conjugate of ${H^1_{\iota_0}}$. The direct sum

$\displaystyle \bigoplus_\iota H^1_\iota$

is defined over ${{\mathbb Q}}$.

This gives rise to a bundle of abelian varieties

$\displaystyle Jac(\oplus_\iota H^1_\iota){\rightarrow} X$

Theorem 1 (Moeller) For any ${x\in X}$ and any ${p,q}$ zeroes of ${\omega_x}$ on ${C_x}$ we have that

$\displaystyle [p-q]\in Jac(\oplus_\iota H^1_\iota)_x$

is a torsion point. We view ${[p-q]}$ as a divisor of degree zero, which then maps to the Jacobian by the Abel-Jacobi map. A different way to state the theorem is that for any holomorphic ${1}$-form in ${\oplus_\iota H^1_\iota}$ the relative period from ${p}$ to ${q}$ is a rational combination of the absolute periods of the same ${1}$-form.

— 2. Properties of ${C{\rightarrow} X}$

To prove the above theorem, Moeller uses the notion of a “maximal Higgs bundle”. This implies vanishing of a cohomology group associated to the local system ${H^1_{\iota_0}}$ and a similar vanishing follows for the Galois conjugates. The point ${[p-q]}$ would give rise to a non-trivial cohomology class above, unless it was torsion.

The proof below is different and uses the embedding ${X{\hookrightarrow} {\mathcal H}/{\mathbb C}^\times}$. This gives rise to an exact sequence of bundles over ${X}$

$\displaystyle 0{\rightarrow} W_0 {\rightarrow} T{\mathcal H} {\rightarrow} H^1{\rightarrow} 0$

Here is the notation. By ${T{\mathcal H}}$ we denote the tangent space to ${{\mathcal H}}$, which is the same as a relative cohomology group of ${C_x}$ for ${x\in X}$. By ${W_0}$ we denote the purely relative part, which corresponds to the zeroes of ${\omega_x}$. By ${H^1}$ we denote the absolute cohomology ${H^1(C_x)}$.

The proof goes by considering the position of the tangent space of ${X}$ inside ${{\mathcal H}}$, i.e. of the inclusion ${TX\subseteq T{\mathcal H}}$. Considering the projection ${p:T{\mathcal H}{\rightarrow} H^1}$, by work of Moeller we have ${p(TX)=H^1_{\iota_0}}$.

The next step is to understand the position of ${TX}$ relative to the ${W_0}$ piece of ${T{\mathcal H}}$. Recall that we have the bundle of holomorphic ${1}$-forms ${F^1\subseteq H^1}$. This can be lifted to ${F^1\subseteq T{\mathcal H}}$ since a holomorphic ${1}$-form gives also a relative cohomology class.

The following example is taken almost verbatim from the paper. It explains in a simple situation how a point on a torus describes the relative position of a subspace.

Example 2 Consider ${H^1_{\mathbb Z}=\langle a,b\rangle}$ with filtration ${F^1 H^1=\langle a+\tau b\rangle}$, where ${Im( \tau)>0}$. Consider possible extensions of the form

$\displaystyle 0{\rightarrow} W_0{\rightarrow} E {\rightarrow} H^1{\rightarrow} 0$

Let ${W_0}$ be of ${{\mathbb Z}}$-rank ${1}$, generated by ${c}$. Fix some ${{\mathbb Z}}$-lift to ${E}$ of the generators ${a,b}$ of ${H^1}$, denoted ${a_1,b_1}$. Then the data of a lift of ${F^1}$ to ${E}$ is some vector of the form ${x=\mu c + a_1 + \tau b_1}$. Note that we could have chosen different lifts of ${a,b}$ to ${E}$, by adding any integer multiple of ${c}$. This would change ${\mu}$ by integer multiples of ${1}$ and ${\tau}$, i.e. ${\mu}$ is well-defined in ${{\mathbb C}/\langle 1,\tau\rangle}$. This is a point on an elliptic curve, and it is torsion if and only if ${F^1\subseteq E}$ intersects the ${{\mathbb Z}}$-lattice of ${E}$.

With this example in mind, we can now state the necessary result.

Proposition 3 (Main Proposition) With the notation as in the beginning of the section we have

$\displaystyle TX\cap F^1 = \langle\omega\rangle$

inside ${T{\mathcal H}}$, not just after projection to ${H^1}$.

Remark 4 The statement above might seem trivial, and indeed for the tautological bundle it is. But the same proof will work for the Galois conjugates of ${TX}$ inside ${T{\mathcal H}}$, denote them ${TX_\iota}$. One needs to replace ${\omega}$ by ${F^1_\iota}$, which is the holomorphic ${1}$-form corresponding to the factor ${H^1_\iota}$. One can then piece together the results for each ${\iota}$. Reasoning as in the example above, this implies that the points ${[p-q]}$ must be torsion. Note the proposition above says that the transcendental subspace ${F^1}$ must intersect the subspace ${TX}$, defined over a number field. This can be viewed as a linear restriction on the relative periods of ${\omega}$.

— 3. Proof of the Main Proposition —

Consider the short exact sequence

$\displaystyle 0{\rightarrow} W_0 {\rightarrow} T{\mathcal H} {\rightarrow} H^1 {\rightarrow} 0$

Denote the right projection map by ${p:T{\mathcal H}{\rightarrow} H^1}$. Inside ${T{\mathcal H}}$ we have the tangent space to ${X}$ denoted ${TX}$. It maps isomorphically to ${H^1_{\iota_0}}$ via the map ${p}$.

Inside ${H^1_{\iota_0}}$ we have the holomorphic ${1}$-forms ${H^{1,0}_{\iota_0}}$. These also give relative cohomology classes, so they lift to ${T{\mathcal H}}$; denote the subspace by ${F^1_{\iota_0}}$.

Because ${TX}$ maps isomorphically to ${H^1_{\iota_0}}$ via ${p}$, we have two apriori different splittings:

$\displaystyle \begin{array}{rcl} \sigma_{\mathbb R}&:H^{1,0}_{\iota_0} {\rightarrow} F^1_{\iota_0}\subseteq T{\mathcal H}\\ \sigma_{\mathbb Z}&:H^{1,0}_{\iota_0} {\rightarrow} TX \subseteq T{\mathcal H} \end{array}$

We can take the difference of the above maps and compose with the projection ${p}$. By construction, we have

$\displaystyle p\circ (\sigma_{\mathbb R}-\sigma_{\mathbb Z})= 0$

This means that ${\sigma_{\mathbb R}-\sigma_{\mathbb Z}}$ lands in the kernel of ${p}$, which is ${W_0}$. We thus obtain a map

$\displaystyle \sigma_{\mathbb R}-\sigma_{\mathbb Z}:H^{1,0}_{\iota_0}{\rightarrow} W_0$

Now we can pass to a finite cover of ${X}$ where the zeroes of the ${1}$-forms are labeled, i.e. where ${W_0}$ is a trivial local system. Pick a (complex) linear function on ${W_0}$ and compose with the above map to get

$\displaystyle \phi:H^{1,0}_{\iota_0} {\rightarrow} {\mathbb C}$

This map is holomorphic and is a global section of the dual bundle, i.e. of ${H^{0,1}_{\iota_0}}$. This bundle has negative curvature and so cannot have global holomorphic sections. We conclude ${\phi=0}$ and so ${\sigma_{\mathbb R}=\sigma_{\mathbb Z}}$.

This completes the proof.

Remark 5 In the above proof, one needs to check that the behavior of ${|\phi|}$ at the cusps of ${X}$ is not too bad. This can be done, by controlling ${|\phi|}$ along a.e. Teichmuller geodesic. Then, the same technique as in this paper gives that ${\phi}$ must vanish.

— 4. Affine Manifolds —

The proof for general affine manifolds is similar. The decomposition needed is proved in this paper using results of Alex Wright.

The linear algebra is similar, but one can no longer make the claim about all combinations of zeroes. Concretely, if ${{\mathcal M}}$ is an affine manifold and ${T{\mathcal M}}$ is its tangent space, then let

$\displaystyle W_0 T{\mathcal M}:= T{\mathcal M}\cap W_0$

The smaller this space is, the stronger the theorem on torsion is. For Teichmuller curves, the space is empty and the theorem is strongest. For strata, the space is all of ${W_0}$ and the torsion theorem is vacuous.

The space ${W_0T{\mathcal M}}$ is in the purely relative cohomology and its annihilator ${\check{W_0}T{\mathcal M}}$ is in homology. The bigger the space ${W_0T{\mathcal M}}$, the smaller its annihilator (and vice-versa).

Apriori, ${\check{W_0}T{\mathcal M}}$ is a linear subspace defined over the number field ${k\subseteq {\mathbb R}}$. If say ${k={\mathbb Q}(\sqrt{D})}$, then a typical element will look like

$\displaystyle (1+\sqrt{D})[p] + (1-\sqrt{D})[q] - 2[r]$

Here ${p,q,r}$ are zeroes of the ${1}$-form and the above is a ${k}$-linear combination of them which has degree zero. To map such a combination to the Jacobian, rewrite it as

$\displaystyle (1+\sqrt{D})[p-r] + (1-\sqrt{D})[q-r]$

Now each of ${[p-r]}$ and ${[q-r]}$ are divisors of degree zero and so map to the Jacobian. But the Jacobian also has real multiplication by the totally real field ${k}$. In this example, we can view ${\sqrt{D}}$ as an operator on the Jacobian and the above expression provides a point on the Jacobian.

The above construction has an interpretation in terms of relative periods of ${1}$-forms. Pick some cycle ${[r]\in \check{W_0}T{\mathcal M}}$ as above. Then there exists a ${k}$-linear expression ${[a]}$ in absolute homology with the following property. For any holomorphic ${1}$-form ${\omega_\iota}$ in some factor ${H^1_\iota}$ the relative period of ${\omega_\iota}$ on ${[r]}$ equals the absolute period on the Galois conjugate of ${[a]}$.

Namely, ${[a]}$ is a ${k}$-linear combination of absolute ${{\mathbb Z}}$-cycles. Its Galois conjugate is another such object, and we can integrate ${\omega_\iota}$ on the ${{\mathbb Z}}$-cycles and multiply by the ${k}$-coefficients.

Question 1 It would be interesting to see if the above-described situation can actually occur. In all examples I am familiar with, the space ${W_0T{\mathcal M}}$ is actually defined over ${{\mathbb Q}}$. In such a case Galois-conjugation becomes unnecessary, as does the real-multiplication action on the Jacobian.