European Journal of Mathematics

, Volume 4, Issue 3, pp 827–836 | Cite as

A canonical form for a symplectic involution

  • Harry W. Braden
Open Access
Research Article


We present a canonical form for a symplectic involution Open image in new window , \(S^2=\mathop {\mathrm{Id}}\nolimits \); the construction is algorithmic. Application is made in the Riemann surface setting.


Canonical form Symplectic involution 

Mathematics Subject Classification

15A21 14H37 14Q05 

1 Introduction

Canonical forms for matrices with integer coefficients are useful in many settings: one may think of the Smith Normal Form or Frobenius’s decomposition of a skew matrix [15], both of which will be used later in the paper. In his study of real abelian varieties Comessatti [5, 6] introduced a canonical form for an involution in Open image in new window (the precise result will be recalled later). Here we shall establish the symplectic analogue of Comessatti’s theorem providing a canonical form for a symplectic involution, Open image in new window , where throughout \(J= \Bigl ({\begin{matrix}0&{}1_g\\ -1_g&{}0\end{matrix}}\Bigr )\) is the canonical symplectic pairing. The canonical form with an immediate corollary is given by:

Theorem 1.1

Let Open image in new window be a symplectic involution, Open image in new window and Open image in new window . Then S is symplectically equivalent to one of the form \(S=\bigl ({\begin{matrix}a&{}0\\ 0&{}a\end{matrix}}\bigr )\) whereIf t is the number of Open image in new window blocks \(\mathfrak {Q}\) then \(g=p+m+2t\).

Corollary 1.2

Let S be a symplectic involution of \(W=\mathbb {Z}^{2g}\) with canonical pairing. Then Open image in new window , \(\langle L_i,L_i\rangle =0\), with stable Lagrangian subspaces \(SL_i=L_i\).

Some special cases of Theorem 1.1 are known in the context of Riemann surfaces. The canonical form yields a different proof of

Theorem 1.3

Let S be a conformal involution of the Riemann surface \({\mathscr {C}}\) of genus g with k fixed points and let Open image in new window be the quotient surface of genus \(g'\). We may find a homology basis for \({\mathscr {C}}\) in which S takes the form (1) where \( g=p+m+2t\), Open image in new window and either

The former case yields a result of Gilman [12] while the latter yields Fay’s example of an unbranched covers [11, Chapter IV].

We note that the proof we present of Theorem 1.1 is constructive. Before turning to the proofs we will give some further background including Comessatti’s result that we will employ. If one could find a module-theoretic proof of Corollary 1.2 then Theorem 1.1 would follow from Comessatti’s theorem.

Finally we remark that the theorems described in this paper are of significant utility in the computational study of Riemann surfaces. Relevant for this volume Edge often studied curves and geometric configurations with high symmetry such as Klein’s curve [8], Bring’s curve [10] and the Fricke–Macbeath curve [9]. The first and third of these curves are Hurwitz surfaces whose automorphism groups (Hurwitz groups) are quotients of the (2, 3, 7)-triangle group possessing the symplectic involutions of the theorem. Bring’s curve with automorphism group \(S_5\) is a quotient of the (2, 4, 5)-triangle group [17]. Using the theorems outlined in this paper for curves with symmetries adapted homology bases may be found that, for example, significantly simplify the period matrices and the calculation of the vector of Riemann constants [2, 3].

2 Background

In order to place the result in context its helpful to see the parallel between several results for the general linear and symplectic groups. First,

Lemma 2.1 is classical (see for example [15]). Lemma 2.2 seems less well-known; the first proof of this I am aware of is [16].

In his study of real abelian varieties Comessatti [5, 6] introduced the following canonical form.

Theorem 2.3

Let M be a free \(\mathbb {Z}\)-module of rank m and let \(S\in {\mathrm{Aut}}\,(M)\) be an involution. Let Open image in new window where \(M_\pm \) are the submodulesThen we may find a basis of M such that S takes the formwhere we have \(\lambda \) copies of the matrix \(\mathfrak {Q}=\bigl ({\begin{matrix}0&{}1\\ 1&{}0\end{matrix}}\bigr )\). Moreover, \(s_+\), \(s_-\) and \(\lambda \) are invariants of S.
Here \(\lambda \) is known as the Comessatti character (a modern review of Comessatti’s work may be found in [4]). Silhol [18] expressed these invariants in terms of the group cohomology of \(G=\langle 1,S\rangle \). Then
$$\begin{aligned} H^{i}(G,M)={\left\{ \begin{array}{ll} \,M_+ \cong (\mathbb {Z}_2)^{s^+} &{}\quad i=0,\\ \,\dfrac{M_-}{(1-S)M}\cong (\mathbb {Z}_2)^{s^--\lambda }&{}\quad i\equiv 1 \;\mathrm{mod}\; 2,\\ \,\dfrac{M_+}{(1+S)M}\cong (\mathbb {Z}_2)^{s^+-\lambda }&{}\quad i\equiv 0 \;\mathrm{mod}\; 2,\;\; i>0. \end{array}\right. } \end{aligned}$$
In the study of real structures the focus of attention are anti-holomorphic involutions Open image in new window , \(S^2=\mathop {\mathrm{Id}}\nolimits \) rather than holomorphic involutions. Comessatti showed that an anti-holomorphic involution S takes the form \(\Bigl ({\begin{matrix}1_{g}&{}H\\ 0&{}-1_{g}\end{matrix}}\Bigr )\) where H is a symmetric bilinear form over \(\mathbb {Z}_2\). These forms are determined by the rank of H and whether Open image in new window is nonzero or not. We have either (see for example [19])An algorithm that constructs such a basis for a Riemann surface with real structure is given in [14].

Remark 2.4

Comessatti’s theorem admits both purely module-theoretic and constructive proofs. The proof that follows of Theorem 1.1 is constructive. If one could find a module-theoretic proof of Corollary 1.2 then the theorem would follow from Comessatti’s theorem. A generalisation of Comessatti’s theorem exists for all cyclic groups of prime order [7, Theorem 74.3] though the corresponding generalisation of Theorem 1.1 has yet to be established.

3 Proof of Theorem 1.1

The proof is constructive. Writing \(S=\bigl ({\begin{matrix} a &{} b \\ c&{} d \end{matrix}}\bigr )\) where abcd are block Open image in new window integer matrices, the constraints Open image in new window and Open image in new window mean that S takes the formWe remark that if Open image in new window and \(\mu =\mu ^{T}\) then the rotations and translations
$$\begin{aligned} R_U=\begin{pmatrix} U&{} 0\\ 0&{} U^{-1\, T} \end{pmatrix}, \quad T_\mu =\begin{pmatrix} 1&{} \mu \\ 0&{} 1 \end{pmatrix} \end{aligned}$$
are symplectic. In particular the similarity transformationyields a similarity transformation on a and takes c to a congruent matrix.
The first step of the proof is to make a symplectic transformation so that \(c=0\). The Frobenius decomposition of the skew matrix c says there exists \(U\in \mathrm{GL}(g,\mathbb {Z})\) such that \(c=U^{T}\!DU\) where the congruent matrix D takes the formwith Open image in new window for \(1 \leqslant i \leqslant s-1\) and \(\mathop {\mathrm{rank}}\nolimits \, c=2s\). By using the appropriate symplectic transformation \(R_U\) we may suppose that c is in the Frobenius form D stated. Let Open image in new window . Then there are \(p,q,u,v\in \mathbb {Z}\) such that
$$\begin{aligned} a_{21}=\nu p,\quad d_1=\nu q,\quad up-vq=1. \end{aligned}$$
Then the symplectic matrix
$$\begin{aligned} T=\begin{pmatrix} 1&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad u&{} \quad 0&{} \quad 0&{} \quad v&{} \quad 0\\ 0&{} \quad 0&{} \quad 1_{g-2}&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad 0\\ 0&{} \quad q&{} \quad 0&{} \quad 0&{} \quad p&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 1_{g-2} \end{pmatrix}, \end{aligned}$$
is such thatContinuing in this way we see that S is similar via a symplectic transformation to the case when \(c=0\).
With \(c=0\) we see from from (2) that Open image in new window . Using the freedom to make a similarity transform to a, noted above, we may now use Comessatti’s Theorem 2.3 to put a into the canonical formfor appropriate r and s. At this stage we have that
$$\begin{aligned} S=\begin{pmatrix} a &{} b \\ 0&{} a^{T} \end{pmatrix},\quad 0=b+b^{T}\!=ab+ba^{T} \end{aligned}$$
and in block formwhere Q is a Open image in new window matrix. Now solving for Open image in new window we find that b has the form
$$\begin{aligned} b=\begin{pmatrix}0&{} x&{} y\\ {-}x^{T}&{} 0&{} z\\ {-}y^{T}&{} {-}z^{T} &{} \gamma \end{pmatrix},\quad \gamma +\gamma ^{T}\!=0=\gamma Q+Q\gamma ,\; \; y={-}yQ,\;\; z=zQ. \end{aligned}$$
Here \(x\in M_{r,s}(\mathbb {Z})\), \(y\in M_{r,2l}(\mathbb {Z})\), \(z\in M_{s,2l}(\mathbb {Z})\), \(\gamma \in M_{2l,2l}(\mathbb {Z})\). Thus each row of the matrix y takes the formwhile each row of the matrix z takes the formFurther the skew-symmetric matrix \(\gamma \) may be written as Open image in new window blocksObserve thatand so if
$$\begin{aligned} \mu =\begin{pmatrix} \mu _1&{}\mu _2&{} \mu _3\\ \mu _2^{T}&{} \mu _4&{} \mu _5\\ \mu _3^{T}&{} \mu _5^{T}&{} \mu _6 \end{pmatrix},\quad \mu _1=\mu _1^{T},\;\; \mu _4=\mu _4^{T},\;\; \mu _6=\mu _6^{T} \end{aligned}$$
then \(\mu =\mu ^{T}\) andThus if we choose the rows of the matrix \(\mu _3\) to be \((y_{i1},0,y_{i2},0,\ldots ,y_{il},0)\), \( 1\leqslant i\leqslant r\), then
$$\begin{aligned} y+\mu _3Q-\mu _3=0. \end{aligned}$$
Similarly if the rows of the matrix of the matrix \(\mu _5\) to be Open image in new window , \( 1\leqslant j\leqslant s\), then
$$\begin{aligned} z+\mu _5 Q+\mu _5=0. \end{aligned}$$
Finally taking \(\mu _6\) to be of the formyields
$$\begin{aligned} \gamma +\mu _6 Q-Q\mu _6=0. \end{aligned}$$
Therefore we may take b to be of the form formand where x is a (0, 1)-matrix.
At this stage we have shown that we may choose a symplectic basis in which the involution S takes the block formwhere x is a (0, 1)-matrix. Further, by use of the rotation \(R_U\) with U of the formwe may transform x to \(AxB^{T}\). Making use of the Smith normal form and the ability to remove even integral parts of x by a translation we may therefore assume x to have only 1’s and 0’s along the diagonal and be zero off the diagonal. Suppose there are Open image in new window 1’s on the diagonal. Then we may writewhere we have t copies of the symplectic matrixand we are indicating a symplectic decomposition. Now considerThenThus by conjugation we may bring S to the desired form and have established the theorem.

4 Proof of Theorem 1.3

We now apply our results in the setting where we have a Riemann surface \({\mathscr {C}}\) of genus \( g>0\) with nontrivial finite group of symmetries \(G\leqslant {\mathrm{Aut}}\,{\mathscr {C}}\) (\({\mathrm{Aut}}\,{\mathscr {C}}\) is necessarily finite for \(g\geqslant 2\)). \({\mathrm{Aut}}\,{\mathscr {C}}\) acts naturally on \({\mathscr {C}}\), \(H_1( {\mathscr {C}},\mathbb {Z})\) and the harmonic differentials. Consider the quotient Riemann surface Open image in new window of genus \(g'\). From (1) \(g=(p+t)+(m+t)\) and we can form \(p+t\) invariant differentials and \(m+t\) anti-invariant differentials under the action of S; then \(g'=p+t\) is the genus of \(\mathscr {C}'\). By Riemann–Hurwitz if there are \(k\geqslant 0\) fixed points of S then \( g-1 =2(g'-1)+k/2\) yields
$$\begin{aligned} m=p+\frac{k}{2}-1. \end{aligned}$$
Hurwitz showed that \(\phi \in {\mathrm{Aut}}\,\mathscr {C}\) is the identity if and only if it induces the identity on \(H_1({\mathscr {C}},\mathbb {Z})\). Accola [1] strengthened this result and showed that for \(g\geqslant 2\) if there exist two pairs of canonical cycles such that (in homology) \(\phi ({\mathfrak {a}}_1)={\mathfrak {a}}_1\), \(\phi ({\mathfrak {a}}_2)={\mathfrak {a}}_2\), \(\phi ({\mathfrak {b}}_1)={\mathfrak {b}}_1\) and \(\phi ({\mathfrak {b}}_2)={\mathfrak {b}}_2\) then \(\phi \) is the identity (simpler proofs of this result were obtained by Earle as well as Grothendieck and Serre, see [13]). Accola’s result means in the canonical form for the symplectic involution above we have \(p\leqslant 1\). We will have therefore proven the theorem once we establish

Lemma 4.1

If \(k>0\) then \(p=0\).


Let Open image in new window be a basis for \(H_1({\mathscr {C}},\mathbb {Z})\) ordered so that \(\gamma _a=\mathfrak {a}_a\), \(\gamma _{ g+a}=\mathfrak {b}_a \), \(a=1,\ldots ,g\), are canonically paired, \(\langle \mathfrak {a}_a,\mathfrak {b}_b\rangle =\delta _{ab}\), and the symplectic form is \({J}_{ab}=\langle \gamma _a,\gamma _b\rangle \). Let \({\alpha }_b\) denote a basis of the harmonic forms paired with the homology cycles \(\gamma _a\) by Open image in new window . With the metric on (complexified as necessary) one-forms Open image in new window then we also have thatwhere \(*\) is the Hodge star operator.

If Open image in new window then Open image in new window . Letting Open image in new window , Open image in new window denote the analogous quantities for \(\mathscr {C}'\) we may write Open image in new window and similarly for v.

Suppose that \(p>0\). Then (upon possible relabelling) we have \(S({\mathfrak {a}}_1)={\mathfrak {a}}_1\), \(S({\mathfrak {b}}_1)={\mathfrak {b}}_1\) and \(S^*{\alpha }_1={\alpha }_1\), \(S^*{\alpha }_{ g+1}={\alpha }_{g+1}\). Now \(\pi ^*u\) for Open image in new window span the invariant differentials of \(H^{1}({\mathscr {C}},\mathbb {R})\) and we may find u, v such that \(\pi ^*u={\alpha }_1\), \(\pi ^*v={\alpha }_{ g+1}\). We have thatNow suppose that in addition there exists a fixed point P of S. Thus for all Q,and so consequently (as the path between Q and S(Q) may be arbitrary) \(\int _Q^{S(Q)}{\alpha }_1\in \mathbb {Z}\). But now if \(\gamma '\) is any cycle on \(\mathscr {C}'\) containing the arbitrary point \(\pi (Q)\) this may be lifted to a path in \({\mathscr {C}}\) beginning at Q and ending at \(S^{l}(Q)\) for some l (we may assume this lifted path does not pass through any of the fixed points of S). ThenAs this is true for any path \(\gamma '\) we have Open image in new window with \(n_i\in \mathbb {Z}\) and similarly for Open image in new window with \(m_i\in \mathbb {Z}\). Therefore Open image in new window , but from (3) this is not possible. Thus if \(p>0\) then \(k=0\). \(\square \)


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Authors and Affiliations

  1. 1.Maxwell Institute and School of MathematicsThe University of EdinburghEdinburghUK

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