The Distribution of Weierstrass points on a Compact Riemann Surface

1984 ◽  
Vol 120 (2) ◽  
pp. 317 ◽  
Author(s):  
Amnon Neeman
Keyword(s):  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Weierstrass Points

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).

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.

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)).

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)).

PARABOLIC GENERALIZED MODULAR FORMS AND THEIR CHARACTERS

2009 ◽  
Vol 05 (05) ◽  
pp. 845-857 ◽  
Author(s):  
MARVIN KNOPP ◽  
GEOFFREY MASON
Keyword(s):  
Riemann Surface ◽  
Modular Form ◽  
Finite Index ◽  
Modular Forms ◽  
Modular Group ◽  
Compact Riemann Surface ◽  
Eichler Cohomology

We make a detailed study of the generalized modular forms of weight zero and their associated multiplier systems (characters) on an arbitrary subgroup Γ of finite index in the modular group. Among other things, we show that every generalized divisor on the compact Riemann surface associated to Γ is the divisor of a modular form (with unitary character) which is unique up to scalars. This extends a result of Petersson, and has applications to the Eichler cohomology.

scholarly journals Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles

2020 ◽  
Author(s):  
Eric Schippers ◽  
Mohammad Shirazi ◽  
Wolfgang Staubach
Keyword(s):  
Riemann Surface ◽  
Bergman Space ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Dirichlet Space ◽  
Holomorphic Functions ◽  
Finite Collection ◽  
Approximation Theorems ◽  
Arbitrary Genus ◽  
Simply Connected

Abstract We consider a compact Riemann surface R of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate R into two subsets: a connected Riemann surface $$\Sigma $$ Σ , and the union $$\mathcal {O}$$ O of a finite collection of simply connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on $$\mathcal {O}$$ O to the Bergman space of holomorphic forms on $$\Sigma $$ Σ is an isomorphism onto the exact one-forms, when restricted to the orthogonal complement of the set of forms on all of R. We then apply this to prove versions of the Plemelj–Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on $$\Sigma $$ Σ by elements of Bergman space and Dirichlet space on fixed regions in R containing $$\Sigma $$ Σ .

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