Questions tagged [meromorphic-functions]

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Does a vector over the field of meromorphic functions describe a manifold?

Assume that the variables $\mathbf x=(x_1,...,x_n)$ are coordinates on the solution manifold of a differential equation $\mathbf D(\mathbf x,\dot{\mathbf x},\ldots,\mathbf x^{(\alpha)})=\mathbf 0$ ...
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Finite groups of meromorphic functions [closed]

Which finite groups are isomorphic to groups of meromorphic functions on the whole complex plane under composition?
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6 votes
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Does it follow that $F^{(1)}(z)=F^{(2)}(z)$ for all $z \in \mathbb H$?

Let $\mathbb H= \{z \in \mathbb C\,:\, \textrm{Re}\,(z)\geq 0\}$ and for $j=1,2$ suppose that $F^{(j)}:\mathbb H\to \mathbb H$ is defined via $$ F^{(j)}(z) = \sum_{k=1}^{\infty} \frac{a_k^{(j)}}{z+\...
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4 votes
1 answer
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Relationship between Dolbeault and de Rham cohomology on Riemann surface

A lecturer of mine once ``proved'' the existence of non-constant meromorphic functions on a compact Riemann surface $X$ by using analysis of the Laplacian to decompose the de Rham cohomology group as $...
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2 votes
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Triangulating Riemann surfaces by using non-constant meromorphic functions

Let $X$ be a connected Riemann surface, i.e. $X$ is a one dimensional connected complex manifold (Hausdorff and second-countable as a topological space). The following is a classical result: Theorem (...
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3 votes
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weakly holomorphic modular forms with a simple pole at $\infty$

Let $l$ be a prime. Suppose that $M_0^{!}(\Gamma_0(l))$ donote the space of weakly holomorphic modular forms of weight $0$ for the congruence subgroup $\Gamma_0(l)$. Does there exist a $f\in M_0^{!}(\...
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principal divisor on complex surfaces

Let $X$ be a non compact complex surface non projective and non algebraic, and let $S$ be compact Riemann surface embedded in $X$ ( i mean that $S$ is a compact complex sub variety of $X$ of ...
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4 votes
1 answer
287 views

Rouché's Theorem in complex analysis on the relation of the number of zeros and poles of meromorphic functions in a region [closed]

This question is from my son referenced in my earlier question, Need advice or assistance for son who is in prison. His interest is scattering theory . He asked me to post this question: Hello and ...
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1 answer
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Proof of the analytic Fredholm theorem in Borthwick

I've stumbled across a proof of the analytic Fredholm theorem given in Theorem 6.1 in Spectral Theory of Infinite-Area Hyperbolic Surfaces by David Borthwick (see below). Given the notion of being &...
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7 votes
1 answer
219 views

Indeterminacy locus of meromorphic maps of rigid analytic spaces

Setup. Let $k$ be an algebraically closed field of characteristic zero. Let $X/k$ be a normal variety, and let $Y/k$ be a proper variety. It is well-known that the indeterminacy locus of a rational ...
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Is there any non-normal family $\mathcal{F}$ of meromorphic functions on $|z|<1$ whose each zero has multiplicity $2$ but $\mathcal{F'}$ is normal

It is well known that if a family of meromorphic functions is not normal (a family is said to be normal if each sequence of functions in the family has a subsequence which converges locally uniformly ...
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Dimension of global holomorphic sections of a line bundle

Let $K$ be the canonical line bundle of a compact Riemann surface $M$ of genus $g$. Consider the pull back of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the space ...
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2 votes
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168 views

What is meromorphic differentials like on Riemann Sphere? [closed]

There is a proposition that every meromorphic differential on Riemann Sphere (or $\mathbb{P}^1 = \mathbb{C} \cup \{ \infty \}$) can be written as $f dz$ where $f$ is a meromorphic function on $\mathbb{...
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Some simple algebra of rational functions by André Weil

In André Weil's dissertation, he considers two meromorphic functions $x,y$ on a complex curve.  He assumes every pole of $y$ is a pole of $x$, and its multiplicity as a pole of $y$  is no greater than ...
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On a map between Riemann surfaces of genus $1$

Let $C$ be a compact Riemann surface of genus $1$, and $p\in C$, and $w$ be a local holomorphic coordinate on $C$ near $p$ with $w=0$ at $p$. As usual, for a divisor $D$ denote by $L(D)$ the vector ...
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Coefficients of a special meromorphic function

The problem described below appears elementary, but I can't figure out the answer or find it in the literature. I apologize if I have missed something very basic. Let me begin with considering a ...
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4 votes
1 answer
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Poles of an integral of a meromorphic function with toric poles

Suppose I have a meromorphic function in several variables $f(x_1,\ldots,x_k,y_1,\ldots,y_m)$ and I want to integrate along the torus $T^m$ given by $|y_1|=\cdots=|y_m|=1$. It is not true in general ...
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5 votes
1 answer
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A "prequestion" about meromorphic representations of algebraic groups

In a comment exchange around an answer to Is a group scheme determined by its category of representations? there arose the issue of Tannakian reconstruction for non-affine algebraic groups (e. g. ...
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3 votes
0 answers
108 views

Tilings of the plane and meromorphic functions on the plane

This question has three up-votes on m.s.e. but isn't getting any answers. Every textbook says every doubly-periodic meromorphic function on $\mathbb C$ has a fundamental domain that is a ...
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