A canonical form for symmetric and skew-symmetric extended symplectic modular matrices with applications to Riemann surface theory

Author:
Robert Zarrow

Journal:
Trans. Amer. Math. Soc. **204** (1975), 207-227

MSC:
Primary 32G20; Secondary 15A21

DOI:
https://doi.org/10.1090/S0002-9947-1975-0407324-2

MathSciNet review:
0407324

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Abstract: The (extended) symplectic modular group is the set of all integer matrices such that

We use this canonical form to study two types of Riemann surfaces which are conformally equivalent to their conjugates and obtain characterizations of their period matrices. We also obtain characterizations of the symplectic matrices which the conformal equivalence induces on the first homology group. One type of surface dealt with is the symmetric Riemann surfaces, i.e. those surfaces which have a conjugate holomorphic self-map of order **2**. The other type of surface studied we we call pseudo-symmetric surfaces. These are the hyperelliptic surfaces with the property that the sheet interchange is the square of a conjugate holomorphic automorphism.

**[1]**Lars V. Ahlfors and Leo Sario,*Riemann surfaces*, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR**0114911****[2]**Norman L. Alling and Newcomb Greenleaf,*Foundations of the theory of Klein surfaces*, Lecture Notes in Mathematics, Vol. 219, Springer-Verlag, Berlin-New York, 1971. MR**0333163****[3]**Irving Reiner,*Integral representations of cyclic groups of prime order*, Proc. Amer. Math. Soc.**8**(1957), 142–146. MR**0083493**, https://doi.org/10.1090/S0002-9939-1957-0083493-6**[4]**D. Singerman,*Riemann surfaces which are conformally equivalent to their conjugates*(preprint).**[5]**T. A. Springer,*Galois cohomology of linear algebraic groups*, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 149–158. MR**0210714**

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DOI:
https://doi.org/10.1090/S0002-9947-1975-0407324-2

Article copyright:
© Copyright 1975
American Mathematical Society