A transfer principle: from periods to isoperiodic foliations

Abstract

We classify the possible closures of leaves of the isoperiodic foliation defined on the Hodge bundle over the moduli space of genus $g\ge 2$ curves and prove that the foliation is ergodic on those sets. The results derive from the connectedness properties of the fibers of the period map defined on the Torelli cover of the moduli space. Some consequences on the topology of Hurwitz spaces of primitive branched coverings over elliptic curves are also obtained. To prove the results we develop the theory of augmented Torelli space, the branched Torelli cover of the Deligne–Mumford compactification of the moduli space of curves.

Appendices

Appendix I: Proof of Proposition 3.10

We first begin by providing a normal form for periods of positive volume and finite primitive degree, which implies the first item of Proposition 3.10 in the case where the subspace (18) is rational.

Lemma 9.1

Given a surjective homomorphism $p:{\mathbb {Z}}^{2g}\rightarrow {\mathbb {Z}}+ i {\mathbb {Z}}$ of volume (and primitive degree) $d\ge 2$, there exists $M\in \text {Sp}(2g,{\mathbb {Z}})$ and a symplectic basis $\{a_1,b_1,\ldots , a_g,b_g\}$ of ${\mathbb {Z}}^{2g}$ such that

$$\begin{aligned} p\circ M=a_1^*+i(db_1^*+a_2^*) \end{aligned}$$

where the star denotes the symplectic dual of the given element.

Proof

Denote by $x=\Re (p)$ and $y=\Im (p)$ the elements in $({\mathbb {Z}}^{2g})^*$. They satisfy $x\cdot y=d$. Choose $a_1\in {\mathbb {Z}}^{2g}$ the symplectic dual of x. Choose some $b_1\in {\mathbb {Z}}^{2g}$ such that $a_1\cdot b_1=1$ and write

$$\begin{aligned} y=m_1a_1^*+db_1^{*}+m_2a_2^{*} \end{aligned}$$

(58)

where $a_2$ is a primitive element satisfying $a_1\cdot a_2=b_1\cdot a_2=0$ and $m_1,d$ and $m_2$ are co-prime. Complete those elements to a symplectic basis $a_1,b_1, \ldots , a_g,b_g$ of ${\mathbb {Z}}^{2g}$. The image of an element $u=\sum _{k\ge 1}\alpha _ka_k+\beta _kb_k$ under p is

$$\begin{aligned} p(u)=(a_{1}^{*}(u), (m_1a_1^*+db_1^{*}+m_2a_2^{*})(u))=(\alpha _1, m_1\alpha _1+d\beta _1+m_2\alpha _2). \end{aligned}$$

Therefore

$$\begin{aligned} p(u)= & {} 1\Leftrightarrow \alpha _1=1\text { and } m_1+d\beta _2+m_2\alpha _2=0, \\ p(u)= & {} i\Leftrightarrow \alpha _1=0\text { and } d\beta _1+m_2\alpha _2=1. \end{aligned}$$

Hence, p is surjective if and only if d and $m_2$ are co-prime.

To conclude, we are going to show that under the hypothesis of d and $m_2$ co-prime, there exists a choice of $b_1$ such that the coefficients $m_1$ and $m_2$ in the splitting given by (58) are 0 and 1 respectively.

Let us analyze the effect of a change of the first given $b_1$. Write $\widetilde{a_1}^*=a_1^*$ and

$$\begin{aligned} \widetilde{b_1}^*=u_1a_1^*+b_1^*+\sum _{k\ge 2}u_ka_k^*+v_kb_k^* \end{aligned}$$

for some $u_k,v_k\in {\mathbb {Z}}$. The new decomposition $y=\widetilde{m_1}\widetilde{a_1}^*+d\widetilde{b_1}^*+\widetilde{m_2}\widetilde{a_2}^*$ has the properties

$$\begin{aligned} (y-\widetilde{m_1}\widetilde{a_1}^*-d\widetilde{b_1}^*)\cdot \widetilde{a_1}^*=0\text { and } (y-\widetilde{m_1}\widetilde{a_1}^*-d\widetilde{b_1}^*)\cdot \widetilde{b_1}^*=0. \end{aligned}$$

The first equation is automatically satisfied, and the second gives

$$\begin{aligned} \widetilde{m_1}=m_1-du_1+m_2v_2. \end{aligned}$$

(59)

Since d and $m_2$ are co-prime we can already choose $u_1$ and $v_2$ to get $\widetilde{m_1}=0$ and restart the argument by supposing $m_1=0$.

For this choice, we have

$$\begin{aligned} \widetilde{m_2}^*\widetilde{a_2}^*=y-\widetilde{m_1}\widetilde{a_1}^*-d\widetilde{b_1}^*=m_2v_2 a_1^{*}+(m_2-du_2)a_2^*-dv_2b_2^*-d(\sum _{k\ge 3}u_ka_k^*+v_kb_k^*) \end{aligned}$$

and from this

$$\begin{aligned} \widetilde{m_2}=\text {gcd}(-m_2v_2, m_2-du_2, -dv_2, -du_3, -dv_3, \ldots , -d u_g, -dv_g). \end{aligned}$$

If we still want $\widetilde{m_1}=0$ we need to impose $m_2v_2=du_1$ by equation (59). The choice $v_2=d$, $u_1=m_2$ and all other coefficients equal to zero gives $\widetilde{m_1}=0$ and $\widetilde{m_2}=1$, as desired. $\square$

We continue the proof of Proposition 3.10, which is reminiscent of Ratner’s theory.

Equipp $\mathbb R^{2g}$ with its canonical symplectic form $\omega (x,y) = \sum _{1\le k\le g} x_{2k}y_{2k+1}-x_{2k+1}y_{2k}$. The volume of a period $p\in \mathbb C^{2g}$ is the symplectic product $V(p)= \omega (\Re p , \Im p )$. Since the action of $\Gamma$ is linear, and that the volume is multiplicative, namely $V(\lambda p ) = |\lambda |^2 V(p)$ for every $\lambda \in \mathbb C$ and $p\in \mathbb C^{2g}$, we can restrict our attention to the action of $\Gamma$ on the subset $X\subset \mathbb C^{2g}$ whose elements have volume 1. In real and imaginary coordinates the set of periods of volume $1$ is then the set of pairs $(x,y)\in {\mathbb {R}}^{2g}\times {\mathbb {R}}^{2g}$ such that $\omega (x,y)=1$.

The simple real Lie group $G = \text {Sp}(2g, \mathbb R)$ acts transitively on the set of couples $(x,y)\in (\mathbb R^{2g})^2$ such that $\omega (x,y)=1$, and that the stabilizer of the couple

$$\begin{aligned} \big ( (1,0,\ldots ,0), (0,1,0,\ldots ,0) \big )\text { is the group } \left( \begin{array}{ccc} 1 &{} &{} \\ &{} 1 &{} \\ &{} &{} \text {Sp}(2g-2,\mathbb R)\end{array} \right) \end{aligned}$$

that we will denote by U in the sequel. Our set X is isomorphic to the homogeneous space G/U. The linear action of $\Gamma$ on X is under the isomorphism $X \simeq G/U$ given by left multiplication on G/U.

Since the group G is simple, that U is generated by unipotent elements, and that $\Gamma$ is a lattice in G, Ratner’s theorem [Rat91] tells us that the closure of the $\Gamma$-orbits on X are homogeneous in the following sense

Theorem 9.2

(Ratner). For every $p\in X$ of the form $p = gU$, there exists a closed subgroup H of G containing $U^g= gUg^{-1}$, such that $\Gamma \cap H$ is a lattice in H, and such that $\overline{\Gamma \cdot p} = \Gamma H p$.

Notice that in our situation, we have $U^g = I_{|W} \oplus \text {Sp}(W^\perp ) \simeq \text {Sp} (2g-2,\mathbb R)$ where $W={\mathbb {R}}\Re p+{\mathbb {R}}\Im p \subset {\mathbb {R}}^{2g}$ is the symplectic subspace associated to the volume one $p\in {\mathbb {C}}^2$.

Let $H_0$ be the connected component of H containing the identity: then $\Gamma \cap H_0$ is still a lattice in $H_0$, and $U^g$ is contained in $H_0$.

If $H_0=G$ then $\overline{\Gamma \cdot p}=G$ and we deduce that the orbit closure is dense in X. Since the closure of $\Lambda (p)$ contains all the $\Lambda (q)$ of elements $q\in \overline{\Gamma \cdot p}$, we have $\overline{\Lambda (p)}={\mathbb {C}}$.

If $H_0$ is a proper subgroup of G, Kapovich observes that it falls into two categories

• (Semi-simple case) $H_0$ is of the form $S \oplus \text {Sp} (W^\perp )$, where S is a Lie subgroup of $\text {Sp}(W)$.

• (Non semi-simple case) $H_0$ is not semi-simple and preserves a line $L \subset W$.

The proof of this dichotomy can be found in [Kap20, p. 12], and is based on Dynkin’s classification of maximal connected complex Lie subgroups of $\text {Sp}(2g,\mathbb C)$, see [Dyn52]. Let L be a maximal complex Lie subgroup of $\text {Sp}(2g,\mathbb C)$ which contains $H_0$. If $H_0\ne \text {Sp} (2g,\mathbb R)$, its Zariski closure in the complex domain is a strict subgroup of $\text {Sp}(2g,\mathbb C)$, so it is contained in a maximal complex Lie (strict) subgroup of $\text {Sp}(2g,\mathbb C)$. It satisfies one of the following properties (see [GOV94, Ch. 6, Thm 3.1, 3.2]):

• (1) $L= \text {Sp} (V) \oplus \text {Sp} (V^\perp )$ for some complex symplectic subspace $V\subset \mathbb C^{2g}$,

• (2) L is conjugated to $\text {Sp} (s,\mathbb C) \otimes \text {SO} (t,\mathbb C)$ where $2g = st$, $s\ge 2$, $t\ge 3$, $t\ne 4$ or $t=4$ and $s=2$,

• (3) L preserves a line of $\mathbb C^{2g}$.

Since $H_0$ contains $U^g$, L contains the complexification of $U^g$, which is nothing but $\text {Id}_{W_ \mathbb C}\oplus \text {Sp} (W_ \mathbb C ^\perp )$, where $W_\mathbb C$ denotes the complexification $W \otimes _\mathbb R \mathbb C$ of W. In case (1), the only possibility is that up to permutation of V and $V^\perp$, we have $W_\mathbb C= V$. In particular, $H_0$ is a subgroup of $\text {Sp} (W) \oplus \text {Sp}(W^\perp )$. Since it contains $\text {Id}_{|W} \oplus \text {Sp}(W^\perp )$, it must be of the form $S \oplus \text {Sp} (W^\perp )$, where S is a Lie subgroup of $\text {Sp}(W)$. Case (2) cannot occur. In case (3), observe that the line L needs to be in $W_\mathbb C$, since the group $\text {Id}_{|W_\mathbb C} \oplus \text {Sp} (W_\mathbb C ^\perp )$ preserves this line. If L is defined over the reals, we are done. If not, both L and ${\overline{L}}$ (the image of L by the complex conjugation) are preserved by $H_0$, and thus $H_0$ is a subgroup of $\text {Sp} (W) \oplus \text {Sp} (W^\perp )$. As before, because it contains $\text {Id}_W \oplus \text {Sp} (W^\perp )$, it must be of the form $S \oplus \text {Sp}(W^\perp )$, where S is a Lie subgroup of $\text {Sp}(W)$.

Remark 9.3

If $\Gamma _W:= \Gamma \cap (\text {Id}_W \oplus \text {Sp}(W^\perp ))$ is a lattice in $\text {Id}_W \oplus \text {Sp}(W^\perp )$ then W is defined over ${\mathbb {Q}}$ and the conclusion of the first item of Proposition 3.10 follows easily. Indeed, $\Gamma _W$ acts by the identity on $W^\sigma$ for every Galois automorphism $\sigma$. The Zariski closure of $\Gamma _W$ being $\text {Id}_W \oplus \text {Sp} (W^\perp )$ (by Borel density theorem, see [Zim84]), $\text {Id}_W \oplus \text {Sp}(W^\perp )$ acts by the identity on $W^\sigma$ as well. This implies that $W^\sigma = W$ for every $\sigma$ (otherwise $\text {Id}\oplus \text {Sp}(W^{\perp })$ acts by the identity on the non-trivial subspace $(W+W^{\sigma })\cap W^{\perp }$), and so W is rational.

Let us proceed to analyze the different cases.

Semi-simple case. Since the group $H_0 = S\oplus \text {Sp} (W^\perp )$ contains a lattice, it must be unimodular. In particular, either S is the trivial group, or a 1-parameter subgroup, or the whole $\text {Sp} (W)$. If S is trivial, then Ratner’s Theorem tells us $\Gamma _W:= \Gamma \cap (\text {Id}_W \oplus \text {Sp}(W^\perp ))$ is a lattice in $\text {Id}_W \oplus \text {Sp}(W^\perp )$ and we fall in case (1) by Remark 9.3.

If S is 1-dimensional, $S\oplus \text {Id}_{W^\perp }$ would be the radical of $H_0$, and a theorem of Wolf and Raghunathan, see [Rag72], shows that it would intersect $\Gamma$ in a lattice. This implies that the intersection of $\Gamma$ with $\text {Id}_W \oplus \text {Sp}(W^\perp )$ is also a lattice. Indeed, let ${\overline{\Gamma }}$ denote the natural projection of $\Gamma$ on $\text {Sp}(W^{\perp })$. Since the sequence

$$\begin{aligned} 0\rightarrow S\cap \Gamma \rightarrow \Gamma \rightarrow {\overline{\Gamma }}\rightarrow 1 \end{aligned}$$

is exact, the map $\Gamma \rightarrow (S\cap \Gamma )\backslash S$ induced by the projection of $S\oplus \text {Sp}(W^{\perp })$ to S induces a morphism ${\overline{\Gamma }}\rightarrow (\Gamma \cap S)\backslash S$. The group $(\Gamma \cap S)\backslash S$ is abelian, and since ${\overline{\Gamma }}$ has Kazdhan property (T), the image group in $(\Gamma \cap S)\backslash S$ is finite. We thus conclude that the image of $\Gamma$ on the factor S is a lattice and therefore $\Gamma \cap (\text {Id}_W \oplus \text {Sp}(W^\perp ))$ as well. We conclude as in Remark 9.3.

Finally, it remains to treat the case where $S= \text {Sp}(W)$. This case splits into two subcases, depending on the lattice $\Gamma \cap \text {Sp} (W) \oplus \text {Sp} (W^\perp )$ being reducible or irreducible. If it is reducible, this implies that $\Gamma \cap (\text {Id}_W \oplus \text {Sp} (W^\perp ))$ is a lattice, and then W must be rational by the above considerations. Assume now that we are in the irreducible case. Then $g=2$, by a theorem of Margulis [Mar91]. Assume W is not rational, otherwise we are done. Let $\sigma$ be a Galois automorphism such that $W^\sigma \ne W$. The group $\Gamma \cap \text {Sp} (W) \oplus \text {Sp} (W^\perp )$ preserves the splitting $W^\sigma \oplus (W^\sigma )^\perp$, since $\Gamma$ and the symplectic form are defined over the rationals. Borel density theorem applied to the lattice $\Gamma \cap \text {Sp} (W) \oplus \text {Sp} (W^\perp )$ shows that $\text {Sp}(W) \oplus \text {Sp} (W^\perp )$ preserves the splitting $W^\sigma \oplus (W^\sigma )^\perp$. This implies that $W^\sigma = W^\perp$ and $(W^\sigma )^\sigma = W$. This being true for every Galois automorphism, this means that W is defined over a totally real quadratic field K, and we have $W^\sigma = W^\perp$ where $\sigma$ is the Galois automorphism of K. This is the only situation where we fall in the last case of Proposition 3.10.

Non semi-simple case. We suppose that $H_0$ is not semi-simple and prove in this case that the periods p satisfy the second case of Proposition 3.10. We already know that there is a line $L\subset W$ that is invariant by the action.

For this, we will first need to understand in detail the subgroup B of $\text {Sp}(2g,\mathbb R)$ formed by all elements that stabilize the line L, see [Kap20, p. 10]. To unscrew the structure of B, notice that any element of B stabilizes both L and $L^\perp$ so that we have an exact sequence

$$\begin{aligned} CH_{2g}\rightarrow B \rightarrow \text {Sp} (L^\perp /L)\simeq \text {Sp}(2g-2). \end{aligned}$$

The group $CH_{2g}$ is then the set of elements $M\in \text {Sp} (2g)$ which induce the identity map on $L^\perp /L$.

We now have another exact sequence

$$\begin{aligned} H_{2g-1} \rightarrow CH_{2g} \rightarrow \text {GL}(L) \simeq \mathbb R^*, \end{aligned}$$

(60)

the last arrow being given by the restriction of an element $M\in CH_{2g}$ to the line L. Hence the subgroup $H_{2g-1}\subset CH_{2g}$ is the group of elements $M\in \text {Sp} (2g)$ which act as the identity on L and on $L^\perp / L$. Such M are easily seen to be of the form $M_{\varphi , \alpha }$, for some $\varphi \in (L^\perp /L)^*$ and $\alpha \in \mathbb R$, where

• the restriction of $M_{\varphi , \alpha }$ to $L^\perp$ equals $id_{|L^\perp } + \varphi a_1$,

• $M_{\varphi , \alpha } (b_1) = \alpha a_1 + b_1 + \sum _{k\ge 2} \varphi (b_k) a_k -\varphi (a_k) b_k$,

where $a_1,b_1,\ldots , a_g,b_g$ is a symplectic basis such that $L= \mathbb R a_1$. The group structure on $H_{2g-1}$ is then given by the following relation

$$\begin{aligned} M_{\varphi , \alpha } M_{\varphi ' , \alpha '} = M_{\varphi + \varphi ', \alpha + \alpha ' + \omega (\varphi , \varphi ')}, \end{aligned}$$

(61)

where $\omega (\varphi , \varphi ')$ is the natural symplectic product induced by $\omega$ on $(L^\perp /L)^*$, namely

$$\begin{aligned} \omega (\varphi , \varphi ') = \sum _{k\ge 2} \varphi (a_k)\varphi '(b_k)-\varphi '(a_k) \varphi (b_k). \end{aligned}$$

Equation (61) is a straightforward computation. An equivalent formulation is that $H_{2g-1}$ is the central extension

$$\begin{aligned} \mathbb R \rightarrow H_{2g-1} \rightarrow (L^\perp / L)^*, \end{aligned}$$

defined by the 2-cocycle $(\varphi , \varphi ') \mapsto \omega (\varphi , \varphi ')$. The group $H_3$ is isomorphic to the classical Heisenberg group of upper triangular real matrices of size $3\times 3$ with 1’s on the diagonal.

Now $CH_{2g}$ is a semi-direct product of $\mathbb R^*$ by $H_{2g-1}$, see (6073). To understand its structure, we introduce for every $\lambda$, one of its lift $S_\lambda \in CH_{2g}$ defined by

$$\begin{aligned} S_\lambda (a_1) = \lambda a_1, \ S_\lambda (b_1) = \frac{1}{\lambda } b_1, \ S_\lambda (a_k) = a_k,\ S_\lambda (b_k)= b_k\text { for } k\ge 2.\end{aligned}$$

A trivial computation shows that for any $\lambda \in \mathbb R^*$, every $\varphi \in (L^\perp /L)^*$ and every $\alpha \in \mathbb R$, we have

$$\begin{aligned} S_\lambda M_{\varphi , \alpha } S_\lambda ^{-1} = M_{\lambda \varphi , \lambda ^2 \alpha }. \end{aligned}$$

(62)

This shows that $CH_{2g}$ is not unimodular, and consequently does not contain any lattice.

By construction, our group $H_0$ is contained in B. We have an exact sequence $CH_{2g} \rightarrow B \rightarrow \text {Sp}(L^\perp /L, \omega )$. The image of $H_0$ by the right arrow is onto since $H_0$ contains $U^g$, so that $H_0$ itself splits as an exact sequence $CH_{2g} \cap H_0 \rightarrow H_0 \rightarrow \text {Sp} (L^\perp /L, \omega )$. The group $CH_{2g}\cap H_0$ is invariant under the action by conjugation of $\text {Sp} (L^\perp /L, \omega )\simeq \text {Sp}(W^{\perp })$. The restriction of this action on $H_{2g-1}$ can be described explicitly: for $U\in \text {Sp}(L^{\perp }/L)\simeq \text {Sp}(W^{\perp })$ denote $s(U)=Id_W\oplus U\in H_0\subset B$. Then

$$\begin{aligned} s(U)M_{\varphi ,\alpha }s(U)^{-1}=M_{\varphi \circ s(U)^{-1},\alpha }. \end{aligned}$$

Lemma 9.4

The closed non-trivial connected subgroups of $CH_{2g}$ invariant by $\text {Sp}(W^{\perp })$ are

(1):

$Z(H_{2g-1}),$

(2):

lifts of $\text {GL}^+(L)$ in $CH_{2g},$

(3):

$H_{2g-1},$

(4):

$\text {GL}^{+}(L) < imes Z(H_{2g-1}),$

(5):

$CH_{2g}.$

The proof of Lemma 9.4 is an easy consequence of the previous exact sequences and calculations.

Now $H_0\cap CH_{2g}$ cannot fall in cases (1) and (2) of Lemma 9.4 since in either of those, $H_0$ would be semi-simple, contrary to hypothesis. It can neither fall in cases (4) or (5), since in those cases equation (62) does not allow $H_0$ to be unimodular. Hence we are left with the possibility $H_0\cap CH_{2g}=H_{2g-1}$ (and $H_0\simeq \text {Sp}(W^{\perp }) < imes H_{2g-1}$). In this case we will show that the invariant line L is rational, and thus we fall in the second possibility of Proposition 3.10.

The theorem of Raghunathan and Wolf cited above tells us that $\Gamma \cap H_{2g-1}$ is a lattice in $H_{2g-1}$. By using Borel’s density Theorem in [Gro88, p.91] we deduce that its Zariski closure is $H_{2g-1}$. We have $L\subset K:=\bigcap _{\gamma \in \Gamma \cap H_{2g-1} } \text {Ker} (\gamma - I)$ which is an intersection of rational spaces. If the inclusion is proper, then the Zariski closure of $\Gamma \cap H_{2g-1}$ would not be the whole of $H_{2g-1}$. This shows that $L=K$ and it is a rational one-dimensional subspace of $\mathbb R^{2g}$.

Up to a real affine change of coordinates on $\mathbb C$, we can assume that the imaginary part of p generates L, and that it is a primitive element of $\mathbb Z^{2g}$. Since the group $H_{2g-1}$ acts transitively on the set of vectors $v\in \mathbb R^{2g}$ such that $v\cdot \Im p= 1$, while keeping the period $\Im p$ fixed, we see that $H\cdot p$ already contains all the periods q such that $\Im q = \Im p$ and such that $V(q)= V(p)= 1$. Since, $\Gamma$ acts transitively on the set of primitive elements of $\mathbb Z^{2g}$, we infer that $\Gamma H p = \overline{\Gamma \cdot p}$ contains all the periods q with volume $V(p) = 1$ and with a primitive integer imaginary part. Since any periods of $\overline{\Gamma \cdot p}$ is of this form, we deduce that this situation is exactly the second case of the proposition. The proof of this latter is now complete.

Applying Moore’s ergodic theorem in [Moo66] to each case H above we deduce the final ergodicity part of Proposition 3.10.

Appendix II: Proof of Lemma 3.9

Up to composing $p$ by an element of $\text {GL}_2^+ (\mathbb R)$ we can assume that the image of $p$ is the set of Gaussian integers. By Lemma 9.1, we can assume that there exists a symplectic basis $a_1, b_1, \ldots , a_g, b_g$ in which the period $p$ has the following form:

$$\begin{aligned} p(a_1)=p(a_2)=1, \ p(b_1)=p(b_2)=i \text { and } p(a_k)= p(b_k)=0 \text { for } k\ge 3. \end{aligned}$$

This basis permits to identify $H_1(\Sigma _g,\mathbb Z)$ with $\mathbb Z^{2g}$ equipped with the symplectic form

$$\begin{aligned} u\cdot v = u_1 v_2 - u_2 v_1 +\cdots + u_{2g-1} v_{2g} - u_{2g} v_{2g-1}, \end{aligned}$$

and the group $\text {Aut}(H_1(\Sigma _g, \mathbb Z))$ with $\text {Sp}(2g,\mathbb Z)$. In these coordinates we have

$$\begin{aligned} p (u_1, \ldots , u_{2g}) = (u_1+u_3) + (u_2+u_4)i . \end{aligned}$$

The form $u_1+u_3 \in (\mathbb Z^{2g})^*$ is dual to the vector $P_1= -(b_1+b_2)$ (meaning that $u_1+u_3= - (b_1+b_2)\cdot u$), whereas the form $u_2+u_4\in (\mathbb Z^{2g})^*$ is dual to $P_2= a_1+a_2$ (meaning $u_2+u_4= (a_1+a_2)\cdot u$). We then have that an edomorphism of ${\mathbb {Z}}^{2g}$ given by a matrix $M$ stabilizes $p$ if and only if $M (P_k)=P_k$ for $k=1,2$, and similarly an endomorphism of $(\mathbb Z/2\mathbb Z)^{2g}$ given by a matrix $M[2]$ stabilizes p[2] if and only if $M[2] (P_k[2])=P_k[2]$ for $k=1,2$. The condition can be read in the columns of the matrices. If $C_k$ (resp. $C_k[2]$) denotes the kth columun of M (resp. M[2]) then

$$\begin{aligned}{} & {} C_1 +C_3= a_1+a_2 \text { and } C_2 +C_4=b_1+b_2 \end{aligned}$$

(63)

$$\begin{aligned}{} & {} (\text {resp. } C_1[2]+C_3[2]=a_1[2]+a_2[2] \text { and } C_2[2]+C_4[2]=b_1[2]+b_2[2]).\end{aligned}$$

(64)

Let $M[2]\in \text {Stab} _{\text {Sp}(2g,\mathbb Z / 2\mathbb Z)} (p[2])$ a symplectic isomorphism whose columns $C_1[2] , \ldots , C_{2g}[2]$ satisfy (64). Let $C_k' \in \mathbb Z^{2g}$ be representatives of the classes $C_k[2]\in (\mathbb Z / 2\mathbb Z)^{2g}$, for $k=1,\dots , 2g$, and $M'$ be the square matrix whose columns are the $C_k'$. Up to changing the representative in each class, we can suppose (63) is valid for the columns of $M'$. We will assume in the sequel that these equations are always satisfied.

Our goal is to modify the vectors $C_k '$ by defining

$$\begin{aligned} C_k := C_k ' + 2E_k, \text { with } E_k\in \mathbb Z^{2g} \end{aligned}$$

in such a way that the matrix $M:= (C_1,\ldots , C_{2g})$ not only stabilizes p (satisfying (63)) but also belongs to $\text {Sp} (2g,\mathbb Z)$. It is therefore necessary to impose

$$\begin{aligned}{} & {} E_1+E_3=0 \text { and } E_2+E_4=0,\end{aligned}$$

(65)

$$\begin{aligned}{} & {} C_1,\ldots , C_{2g}\text { forms a symplectic basis of } {\mathbb {Z}}^{2g}, \end{aligned}$$

(66)

Main step: construction of $C_1,C_2,C_3,C_4$. The conditions we need to satisfy are

• (1) $C_1\cdot C_2= 1$,

• (2) $C_1\cdot C_3=0$, or equivalently $C_1 \cdot (a_1+a_2) =0$,

• (3) $C_1\cdot C_4= 0$, or equivalently, knowing (1): $C_1\cdot (b_1+b_2)=C_1\cdot C_2 =1$,

• (4) $C_2\cdot C_3=0$, or equivalently, knowing (1): $C_2 \cdot (a_1+a_2)=C_2\cdot C_1=-1$,

• (5) $C_2\cdot C_4=0$, or equivalently, $C_2 \cdot (b_1+b_2) =0$,

• (6) $C_3\cdot C_4=1$, or equivalently, $((a_1+a_2) -C_1)\cdot ((b_1+b_2) - C_2) =1$.

If (63) and (1)–(5) are true, we have that (6) is automatically satisfied

$$\begin{aligned}{} & {} ((a_1+a_2) -C_1)\cdot ((b_1+b_2) - C_2) \\{} & {} \quad =(a_1+a_2)\cdot (b_1+b_2) -(a_1+a_2) \cdot C_2 -C_1 \cdot (b_1+b_2) +C_1\cdot C_2= 2-1-1+1=1. \end{aligned}$$

Let us first find $C_1, C_2, C_3, C_4$ so that conditions (2)–(5) are satisfied. In real and imaginary coordinates this is equivalent to

$$\begin{aligned} p (E_1) = \left( \frac{1-C_1'\cdot (b_1+b_2) }{2}, \frac{C_1' \cdot (a_1+a_2)}{2} \right) , \end{aligned}$$

and

$$\begin{aligned} p(E_2) = \left( \frac{(b_1+b_2)\cdot C_2' }{2} , \frac{1+C_2'\cdot (a_1+a_2)}{2} \right) . \end{aligned}$$

Observe that the right hand sides of the last two equations are integers because $C_k'$’s are lifts of - the elements of the symplectic basis- $C_k[2]$’s and satisfy (63). By surjectivity of $p$ there exist solutions to the equations (2)–(5), and so we can assume that the $C_k'$’s satisfy (2)–(5). We can now replace $C_k'$ by $C_k = C_k'+2E_k$ for $k=1,2$ with

$$\begin{aligned} E_k \in (a_1+a_2)^{\perp } \cap (b_1+b_2) ^\perp \text { for } k=1,2. \end{aligned}$$

to preserve conditions (2)–(5). Condition (1) is equivalent to

$$\begin{aligned} 1= C_1 \cdot (C_2' +2 E_2) = C_1\cdot C_2' + 2 C_1\cdot E_2, \end{aligned}$$

and we know that $C_1 \cdot C_2'$ is odd. So we are done if we can choose $C_1$ so that the map

$$\begin{aligned} E_2 \in (a_1+a_2)^{\perp } \cap (b_1+b_2) ^\perp \mapsto C_1\cdot E_2\in \mathbb Z \end{aligned}$$

(67)

is onto. We have $(a_1+a_2)^{\perp } \cap (b_1+b_2) ^\perp = \mathbb Z (a_1-a_2) +\mathbb Z (b_1-b_2) +\sum _{k\ge 3} \mathbb Z a_k+\mathbb Z b_k$ So the map (67) is onto if and only if

$$\begin{aligned} \text {gcd} ( C_1 \cdot (a_1-a_2) , C_1 \cdot (b_1-b_2) , C_1 \cdot a_3 , C_1\cdot b_3, \ldots , C_1 \cdot a_g , C_1\cdot b_g) =1. \end{aligned}$$

(68)

However, observe the following:

• $C_1 \cdot (b_1-b_2)= C_1 \cdot (b_1+b_2) [2]= 1[2]$ is odd, and

• If $g\ge 3$, choosing $E_1$ appropriately, we can assume that the value of $C_1 \cdot a_3$ is either $1\text { or } 2$.

So (68) is satisfied with these choices of $E_1$. Hence the main step is achieved.

To conclude the proof of Lemma 3.9, we need to construct the columns $C_5,\ldots , C_{2g}$. We will inductively find the pair $C_5, C_6$, then the pair $C_7, C_8$, etc.. Let us construct the first one. We need to find $E_5, E_6$ so that

$$\begin{aligned} C_5, C_6 \in \left( C_1, C_2, C_3, C_4 \right) ^\perp \text { and } C_5\cdot C_6 = 1. \end{aligned}$$

This means that the equations

$$\begin{aligned} (E_k\cdot C_1, \ldots , E_k \cdot C_4) =-\frac{1}{2} (C_k'\cdot C_1,\ldots ,C_k'\cdot C_4) \text { for } k=5,6 \end{aligned}$$

(69)

and

$$\begin{aligned} C_5\cdot (C_6'+ 2 E_6) = 1 \end{aligned}$$

(70)

hold. We can find $E_5$ and $E_6$ so that (69) is satisfied, since $C_1, C_2, C_3, C_4$ is a symplectic family (i.e. $C_1\cdot C_2=C_3\cdot C_4=1$ and other products are zero). So we can assume that $C_5'$ and $C_6'$ belong to the orthogonal of $C_1, C_2, C_3, C_4$, which is a symplectic submodule of $\mathbb Z^{2g}$ isomorphic to $\mathbb Z^{2g-4}$ with the canonical symplectic product. In these coordinates, one of the coordinates of $C_5'$ is odd (since $C_5[2]$, the reduction of $C_5'$ modulo 2, is non zero) so by adding to $C_5$ an even vector of $\mathbb Z^{2g-4}$ one can assume that one of the coordinates of $C_5$ is equal to $1$. Hence, equation (70) being equivalent to $C_5\cdot E_6= \frac{-C_5\cdot C_6'}{2}$ and the product $C_5\cdot C_6$ being even, equation (70) can be solved for some suitable choice of $E_6$. We get $C_5$ and $C_6$ in this way. For the construction of the other pairs by induction, the argument is similar.

The first part of the lemma follows, namely the surjectivity of the map $\text {Stab}_{\text {Aut}(H_1(\Sigma _g, \mathbb Z))}(p) \rightarrow \text {Stab}_{\text {Aut}(H_1(\Sigma _g, \mathbb Z / 2\mathbb Z))}(p[2])$.

For the second part, it suffices to remark that the stabilizer of $p[2]$ in $\text {Aut} (H_1(\Sigma _g, \mathbb Z / 2\mathbb Z))$ is the stabilizer of a pair of non colinear elements of $H^1(\Sigma _g, \mathbb Z / 2\mathbb Z)$ that do not intersect each other. The number of elements in this group is given by the announced formula, by elementary considerations.

Appendix III: A result in Picard–Lefschetz theory

We prove here a result which is a well-known consequence of Picard–Lefschetz theory, but which we cannot find as such in the literature.

Let $h : S\rightarrow C$ be a holomorphic map from a compact complex surface $S$ to a compact complex curve $C$. We assume both $S$ and $C$ are non singular, that the critical points of $h$ are non degenerate, and that the fibers of $h$ are connected.

Lemma 11.1

For every $c\in C$, the sequence

$$\begin{aligned} \pi _1 (h^{-1} (c) ) \rightarrow \pi _1 (S) \rightarrow \pi _1 (C) \end{aligned}$$

(71)

where the map on the left is induced by inclusion and the one on the right by $h_*$, is exact.

Proof

Let us first prove the claim for $c$ a regular value of $h$. To do so, given a lift $w\in h^{-1} (c)$, we will use some useful loops $\tilde{\mu _i}: [0,1] \rightarrow S$ starting and ending at $w$ that do not intersect the critical fibers of $h$.

Denote by $c_i$, $i= 1,\ldots , r$ the critical values of $h$. At any regular point $p_i$ of the curve $h^{-1} (c_i )$, the map $h$ induces a biholomorphism between a compact disc $D_i$ transversal to $h^{-1} (c_i$ at $p_i$ and a neighborhood $\delta _i$ of $c_i$ in $C$. We can assume that the discs $\delta _i := h(D_i)$ are disjoint. Given a point $w_i \in \partial D_i$, let $\gamma _i:[0,1] \rightarrow E{\setminus } \cup _i \text {Int} (\delta _i)$ be a family of paths with origin $c$ and end point $h(w_i)$, having the property of being disjoint appart from their origin. Since $h$ induces a $C^\infty$-fibration with connected fibers from $S {\setminus } \cup _i h^{-1} (c_i )$ to $C {\setminus } \{c_i \}$, given a lift $w\in h^{-1} (c)$, we can lift the path $\gamma _i$ to a path $\tilde{\gamma _i}$ starting at $w$ and ending at $w_i$. We denote by $\mu _i$ the concatenation $\gamma _i \star \partial \delta _i\star \gamma _i^{-1}$ and by $\tilde{\mu _i}$ its lift $\tilde{\gamma _i} \star \partial D_i \star \tilde{\gamma _i}^{-1}$. Notice the following two facts:

• (1) $\tilde{\mu _i}$ is homotopically trivial in $S$,

• (2) the representatives of the loops $\mu _i$ in $\pi _1 (C{\setminus } \{c_i \}, c)$ generate the kernel of the map

  $$\begin{aligned} \pi _1 (C{\setminus } \{z_i \}, c)\rightarrow \pi _1 (C, c) \end{aligned}$$

  (72)

  induced by inclusion.

(The first is clear, and the second comes from the fact that the paths $\gamma _i$ are disjoint appart from their origin.)

We are now in a position to prove the claim in the case of a regular fiber. Consider an element in the kernel of the map $h_\star : \pi _1 (S,w) \rightarrow \pi _1(C, c)$ represented by a loop $\varepsilon : [0,1] \rightarrow S$ starting and ending at $w$. Up to homotopy, we can assume that $\varepsilon$ intersects none of the critical fibers of $h$. The image $h \circ \varepsilon$ is then a loop starting and ending at $c$ which defines an element of the fundamental group of $\pi _1 (C {\setminus } \{ z_i \} , c )$ belonging to the kernel of the map (72). Hence, $h\circ \varepsilon$ is homotopic to a word $W (\mu _1, \ldots , \mu _r)$. Consider the path $\varepsilon ' = \varepsilon \star W (\tilde{\mu _1}, \ldots , \tilde{\mu _r}) ^{-1}$; by construction,

• (i) $\varepsilon '$ is homotopic to $\varepsilon$ is $S$,

• (ii) it intersects none of the critical fibers of $h$,

• (iii) its image $h \circ \varepsilon '$ with fixed extremities $c$ is homotopic in $C{\setminus } \{c_i \}$ to the constant path $c$.

Since the restriction of $h$ induces a locally trivial $C^\infty$ fibration from $S {\setminus } \cup _i h^{-1} (c_i )$ to $C {\setminus } \{c_i \}$—this is a proper submersion, so this is Ehresmann’s fibration theorem—we can lift the homotopy found in (iii) to a homotopy in $S {\setminus } \cup _i h^{-1} (c_i )$ between $\varepsilon '$ and a loop contained in the fiber $h^{-1} ( c)$. This establishes that $\varepsilon$ is homotopic with fixed extremities to a loop contained in the fiber $h^{-1} (c)$, and ends the proof of the claim in the case of a regular fiber.

It remains to prove the claim for the singular fibers of $h$. For each $i$, there exists a neighborhood $U$ of $h^{-1} (c_i)$ that retracts by deformation on $h^{-1} (c_i)$ (see [ACG11, Chapter X, Section 9]); in particular, the images of $\pi _1 (h^{-1} (c_i))$ and $\pi _1 (U)$ in $\pi _1 (S)$ coincide. Since $U$ contains a regular fiber, we know from the prerceeding discussion that the image of $\pi _1 (U)$ in $\pi _1 (S)$ contains $\text {Ker} (h_\star )$. On the other hand, the image of $\pi _1 (h^{-1}(c_i))$ in $\pi _1 (S)$ is contained in $\text {Ker} (h_\star )$ since $h$ is constant on $h^{-1} (c_i)$, so the sequence (71) is exact for the critical value $c=c_i$, and the Lemma follows. $\square$

Corollary 11.2

For every $c\in C$, the sequence

$$\begin{aligned} H_1 (h^{-1} (c),\mathbb Z ) \rightarrow H _1 (S,\mathbb Z) \rightarrow H_1 (C,\mathbb Z) \end{aligned}$$

(73)

where the map on the left is induced by inclusion and the one on the right by $h_*$, is exact.

Proof

This is a consequence of Lemma 11.1 and Hurewicz’ theorem. $\square$