A canonical form for a symplectic involution
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Abstract
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.
Keywords
Canonical form Symplectic involutionMathematics Subject Classification
15A21 14H37 14Q051 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
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

\(k>0\) and \(p=0\) whence Open image in new window and \(m=k/21\),

\(k=0\), \(p=1\) whence Open image in new window and \(m=0\).
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 moduletheoretic 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
Lemma 2.2
Lemma 2.1 is classical (see for example [15]). Lemma 2.2 seems less wellknown; 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
Remark 2.4
Comessatti’s theorem admits both purely moduletheoretic and constructive proofs. The proof that follows of Theorem 1.1 is constructive. If one could find a moduletheoretic 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
4 Proof of Theorem 1.3
Lemma 4.1
If \(k>0\) then \(p=0\).
Proof
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.
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