Compact Riemann Surfaces with Conformal Involutions Author(s): Jane Gilman Source: Proceedings of the American Mathematical Society, Vol. 37, No. 1 (Jan., 1973), pp. 105 -107 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2038716 Accessed: 23/03/2009 15:50 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ams. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact
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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 37, Number 1, January 1973
COMPACT RIEMANN SURFACES WITH CONFORMAL INVOLUTIONS1 JANE GILMAN ABSTRACT. In this paper surfaces which have conformal involutions are characterized by their period matrices.
Let S be a compact Riemann surface of genus g, goO, and j a conformal involution on S with k fixed points, k#O. Let S' be the factor surface and g' the genus of S'. Then g=2g'+k/2-1 by the RiemannHurwitz relation, so that k is even. PROPOSITION1. S has a canonicalhomology basis consistingof homology classes of curves of theform (1)
a,, * * * , ag', j(al),
(2)
bl, * * * , bg,,j(bl), * ,j(bg'),
wvhere j(ci)--ci
andj(di)-.-d,
*
,j(ag,),
Cl,
***vCk12-1,
dl,
*
*41c2-1,
and wthere denotes homologous.
PROOF. Let P1, * * *, Pk be the fixed points of j on S and also their images on S'. Let (a, b) be 2g' curves whose homology classes give a canonical homology basis for S'. Let n, be a path on S' from Pi to Pi+1 for all i which does not intersect any curve in (a, b). Then it can be shown that S is two copies of S' slit along the n, for i odd and pasted together appropriately for some choice of slits ni. If c is any path on S', let j be a lifting to S. Let N1=hi-j(hi) for i=1, 2, * * , k-1. Let d,=Ni if i is even; d1=N1; and define d2i+ =d2i - N2i+linductively. Sincej(Ni) =Ni, j(d,)=-di for all i. Let ci=d2i and di+1=d2i+1. Consider the set: al,*
**,
bl, ,
9,
j(al),
*
bg, j(bl), *
,ja,,Cl,
***
Ck12-1,
,j(bg,), dl,
**
7,1Ck2-1.
Let A x B denote the intersection number of A and B. Since Ni x Ni+1=1 for all i, we have: aixbj=6,j; a,xj(b)=O; iixdj=O; a,xc,=O; a,xIj(ai)=O;bixj(b3)=O and cix dj=6j for all relevant i and j. These Received by the editors January 12, 1972. AMS (MOS) subject classifications (1970). Primary 32G20. 1 This paper contains a part of the author's Ph.D. thesis written at Columbia University under the direction of Professor Lipman Bers whom the author wishes to thank for his help and encouragement. ?q)American Mathematical
105
Society 1973
106
[January
JANE GILMAN
intersection numbers can be used to show that these are 2g homologously independent curves. Therefore, their homology classes must form a canonical homology basis for S. A canonical homology basis of the form in Proposition 1 DEFINITION. will be called a homology basis adapted to the involutionj. Then S has a conformal PROPOSITION 2. Assume further that k04. involutionwith kfixed points if and only if it has a period matrix of theform (I, Z) where B A -C Z= A B K IC -IC A and B are g' x g' symmetric, C is g' x (k/2- 1), and K is (k/2- 1) x wvhere (k/2- 1) symmetric. Further, (I, (A ?B)) is a period matrix for S'. Pick a homology basis adapted to j. Let wi be the differential PROOF. which has period 1 with respect to ai and period zero with respect to all other curves in line 1. Let j* be the map whichj induces on differentials. Then]*(wi) has period 1 with respect toj(ai) and period zero with respect to all other curves on line 1. Let wci be the differential with period 1 with respect to ci and period zero with respect to all other curves in line 1. Form the period matrix of S with respect to these bases and use the fact that fj(x)j*(y)=xy for any x and y to simplify the period matrix to the desired form. Finally since (a, b) is a canonical homology basis for S' and since (wi+j*(wi)) as aj-invariant holomorphic differential defines a differential on S', (I, (A ?B)) is a period matrix for S'. To prove the converse we let J,, , kbe the matrix of the action of j on a homology basis adapted to j if j is a conformal involution with k fixed points. Then Z is of the form shown in Proposition 2 if and only if kg, (Z) =Z, where Jg,k, now acts as an element of the (inhomogeneous) Siegel modular group. Indeed, if Jg, ,k viewed as an element of S1,(g,Z), acts on the homology basis, viewed as a column vector whose elements are arranged in the order of (1) then (2) of Proposition 1, then k
jg =,
O-M
where M is g x g and
M=
Ig,
| ?
?
COMPACT RIEMANN SURFACES
1973] Ig" Ik/2-1
107
being the indicated identity matrices; and then Jg',k(Z) = MZM
from which the form of Z follows. J, k(Z)=Z implies (see [2, p. 28]) S has a conformal involution whose 'if the latter occurs, action on homology is either given by Jg ,k or-Jg'k apply the main result of [1] to conclude that either g=O, contrary to assumption, or k is less than 4. In either case, S has a conformal involution and we can apply either the Atiyah-Singer index theorem or the Lefschetz fixed point formula to show that the number of fixed points is 2-trJg', k=k in the one case and 2-tr(-Jg', k)=4-k in the other case. If k is greater than 4, the first case occurs; k=4 is excluded; and if k=2, whichever case occurs, the involution has 2 fixed points. REFERENCES 1. Robert D. Accola, Automorphisms of Riemann surfaces, J. Analyse Math. 18 (1967), 1-5. MR 35 #4398. 2. H. E. Rauch, A transcendental view of the space of algebraic Riemnannsurfaces, Bull. Amer. Math. Soc. 71 (1965), 1-39. MR 35 #4403. DEPARTMENT NEW
YORK
OF MATHEMATICS,
STATE UNIVERSITY
OF NEW
YORK,
STONY BROOK,
11790
Current address: Department of Mathematics, Rutgers University, Newark Campus, Newark, New Jersey 07102
