An Elementary Proof of a Fixed Point Theorem of J. Lewittes and D. L. McQuillan

1978 ◽  
Vol 21 (1) ◽  
pp. 99-101 ◽  
Author(s):  
Arthur K. Wayman
Keyword(s):  
Fixed Point ◽  
Riemann Surface ◽  
Fixed Point Theorem ◽  
Compact Riemann Surface ◽  
Elementary Proof ◽  
Algebraic Function ◽  
Weierstrass Points ◽  
Algebraic Function Fields ◽  
Genus Formula ◽  
Tame Ramification

In (3), J. Lewittes establishes a connection between the number of fixed points of an automorphism of a compact Riemann surface and Weierstrass points on the surface; Lewittes′ techniques are analytic in nature. In (4), D. L. McQuillan proved the result by purely algebraic methods and extended it to arbitrary algebraic function fields in one variable over algebraically closed ground fields, but with restriction to tamely ramified places. In this paper we will give a different proof of the theorem and show that it is an elementary consequence of the Riemann-Hurwitz relative genus formula. Moreover, we can remove the tame ramification restriction.

A Note on Weierstrass Points

1967 ◽  
Vol 19 ◽  
pp. 268-272 ◽  
Author(s):  
Donald L. McQuillan
Keyword(s):  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Function Fields ◽  
Algebraic Function ◽  
Modular Functions ◽  
Weierstrass Points ◽  
Algebraic Function Fields

In (4) G. Lewittes proved some theorems connecting automorphisms of a compact Riemann surface with the Weierstrass points of the surface, and in (5) he applied these results to elliptic modular functions. We refer the reader to these papers for definitions and details. It is our purpose in this note to point out that these results are of a purely algebraic nature, valid in arbitrary algebraic function fields of one variable over algebraically closed ground fields (with an obvious restriction on the characteristic). We shall also make use of the calculation carried out in (5) to obtain a rather easy extension of a theorem proved in (6, p. 312).

scholarly journals On images and inverse images of Weierstrass points

1978 ◽  
Vol 19 (2) ◽  
pp. 125-128
Author(s):  
R. F. Lax
Keyword(s):  
Riemann Surface ◽  
Vector Bundle ◽  
Compact Riemann Surface ◽  
Holomorphic Vector Bundle ◽  
Compact Complex Manifold ◽  
Trivial Bundle ◽  
Weierstrass Points ◽  
Complex Dimension ◽  
Holomorphic Sections ◽  
Image Position

The classical theory of Weierstrass points on a compact Riemann surface is well-known (see, for example, [3]). Ogawa [6] has defined generalized Weierstrass points. Let Y denote a compact complex manifold of (complex) dimension n. Let E denote a holomorphic vector bundle on Y of rank q. Let Jk(E) (k = 0, 1, …) denote the holomorphic vector bundle of k-jets of E [2, p. 112]. Put rk(E) = rank Jk(E) = q.(n + k)!/n!k!. Suppose that Γ(E), the vector space of global holomorphic sections of E, is of dimension γ(E)>0. Consider the trivial bundle Y × Γ(E) and the mapwhich at a point Q∈Y takes a section of E to its k-jet at Q. Put μ = min(γ(E),rk(E)).

scholarly journals On Schoeneberg's theorem

1973 ◽  
Vol 14 (2) ◽  
pp. 202-204 ◽  
Author(s):  
C. Maclachlan
Keyword(s):  
Fixed Point ◽  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Weierstrass Point ◽  
Sufficient Condition

Let Sbe a compact Riemann surface of genus g ≥ 2 and σ an automorphism (conformal self-homeomorphism) of S of order n. Let S* = S/ « σ« have genus g*. In [5], Schoeneberg gave a sufficient condition that a fixed point P ∈ S of σ should be a Weierstrass point of S, i.e., that Sshould support a function that has a pole of order less than or equal to g at P and is elsewhere regular.

scholarly journals Weil–Petersson class non-overlapping mappings into a Riemann surface

2016 ◽  
Vol 18 (04) ◽  
pp. 1550060 ◽  
Author(s):  
David Radnell ◽  
Eric Schippers ◽  
Wolfgang Staubach
Keyword(s):  
Riemann Surface ◽  
Unit Disk ◽  
Bergman Space ◽  
Compact Riemann Surface ◽  
Conformal Mappings ◽  
Hilbert Manifold ◽  
Genus Formula

For a compact Riemann surface of genus [Formula: see text] with [Formula: see text] punctures, consider the class of [Formula: see text]-tuples of conformal mappings [Formula: see text] of the unit disk each taking [Formula: see text] to a puncture. Assume further that (1) these maps are quasiconformally extendible to [Formula: see text], (2) the pre-Schwarzian of each [Formula: see text] is in the Bergman space, and (3) the images of the closures of the disk do not intersect. We show that the class of such non-overlapping mappings is a complex Hilbert manifold.

On (q,n)-gonal pseudo-real Riemann surfaces

2017 ◽  
Vol 28 (13) ◽  
pp. 1750095 ◽  
Author(s):  
Ewa Tyszkowska
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Prime Order ◽  
Compact Riemann Surface ◽  
Quotient Spaces ◽  
Real Riemann Surfaces ◽  
Genus Formula

A compact Riemann surface [Formula: see text] of genus [Formula: see text] is called pseudo-real if it admits an anticonformal automorphism but no anticonformal involution. In this paper, we study pseudo-real [Formula: see text]-gonal Riemann surfaces of genera greater or equal to two; these surfaces have anticonformal automorphisms of prime order [Formula: see text] such that the quotient spaces have genus [Formula: see text].

Weierstrass Points at the Cusps of Γ0(16p) and Hyperellipticity of Γ0(n)

1971 ◽  
Vol 23 (6) ◽  
pp. 960-968 ◽  
Author(s):  
H. Larcher
Keyword(s):  
Riemann Surface ◽  
Positive Integer ◽  
Modular Group ◽  
Compact Riemann Surface ◽  
Weierstrass Point ◽  
Weierstrass Points ◽  
Linear Fractional Transformations ◽  
Fixed Positive Integer

For a fixed positive integer n we consider the subgroup Γ0(n) of the modular group Γ(l). Γ0(n) consists of all linear fractional transformations L: z → (az + b)/(cz + d) with rational integers a, b, c, d, determinant ad – bc = 1, and c ≡ 0(mod n). If ℋ = {z|z = x + iy, x and y real and y > 0} is the upper half of the z-plane then S0 = S0(n) = ℋ/Γ0(n), properly compactified, is a compact Riemann surface whose genus we denote by g(n). A point P of a Riemann surface S of genus g is called a Weierstrass point if there exists a function on S that has a pole of order α ≦ g at P and is regular everywhere else on S.Lehner and Newman started the search for Weierstrass points of S0 (or, loosely, of Γ0(n)).

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