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IV.—The Differential Equations Associated with the Uniformization of Certain Algebraic Curves*

Published online by Cambridge University Press:  14 February 2012

R. A. Rankin
R. A. Rankin
Affiliation:
Department of Mathematics, Glasgow University
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Synopsis

Every algebraic equation can be uniformized by automorphic functions belonging to a certain group of bilinear transformations. In certain cases, such as for hyperelliptic equations, this group is a subgroup of the monodromic group of a differential equation of the form

where R(z) is a rational function which, in general, contains unknown parameters as coefficients. A conjecture of E. T. Whittaker regarding the values of these parameters for the hyperelliptic case is proved for a wide variety of algebraic equations whose branch points possess certain symmetric properties, and is extended to equations of higher type. In several cases, the uniformizing functions belong to subgroups of the groups of the Riemann-Schwarz triangle functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 65 , Issue 1 , 1958 , pp. 35 - 62
Copyright
Copyright © Royal Society of Edinburgh 1958

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References

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