Tagged Questions

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Why is a meromorphic section without zeros and poles on a compact Riemann surface necessarily a constant? Thank you very much.
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maple plot of Belyi function

I would like to understand how to construct Figure 5 of the paper Composition is a generalized symmetry by Alexander Zvonkin: The hypermap/dessin d'enfant of Figure 4 is while the Belyi function ...
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Writing the roots of a polynomial with varying coefficients as continuous functions?

Consider the monic polynomial $$p_{\zeta}(z) = z^n + a_{n-1}(\zeta)z^{n-1} + \dots + a_0(\zeta),$$ where the $a_{i}$'s are continuous functions defined over $\mathbb{C}$. As is well known, the ...
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non-singular Riemann surface implies irreducible polynomial without connectedness?

Let $$F(w,z) = \sum_{i=0}^n a_i(z)w^{n-i}$$ be a polynomial in $z,w$. Define a Riemann surface as the set $$\Gamma:= \left\{ (z,w)\in \mathbb C^2 \mid F(z,w)=0 \right\}$$ and call it non-singular if ...
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function meromorphic on C

Good evening I have a doubt: let $f$ and $g$ are two functions meromorphic on $\mathbb{C}$ such that $g(w) =f(\frac{1}{w})$. Now g is defined for $w = 0$ (because of all meromorphic $\mathbb{C}$).Can ...
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Base-point-free linear systems (elementary?) property

I'm having troubles solving exercise K on page 167 of the book "Algebraic curves and Riemann surfaces" of Miranda. The question is the following one : Let Q be a base-point-free linear system, let ...
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Applications of Stein spaces in Algebraic Geometry

I want to know where are essential applications of the theory of Stein spaces in algebraic geometry. I heard Cartan's theorem A & B were used in Serre's GAGA, but are there any other applications? ...
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Let $\Lambda=<2\omega_1,2\omega_2>$ be a lattice in $\mathbb{C}$, $C=\mathbb{C}/\Lambda$ an elliptic curve. Let $O(0,0)$ (neutral element of the lattice), and $A \in \mathbb{C}/\Lambda$ point of ...
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Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
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Holomorphic functions on algebraic curves

I have been asked to solve the following problem, but I really need some help... How are the holomorphic functions $f:C\to D$, where $C,D$ are nonsingular algebraic curves of genus 1? I know that I ...
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What's the intuition for the fact that $\mathscr{O}(-k)$ and $\mathscr{O}(k)$ are so different?

maybe this question makes no sense and I just cannot accept the fact that dual the line bundle is different from the respective line bundle itself. Since it looks like that manifolds are more ...
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Show that $f(z)= \frac{-1}{z}$ maps each circle of the form $|z+ti| = (t^2-1)^{1/2}$ onto itself.

Show that $f(z)= \frac{-1}{z}$ maps each circle of the form $|z+ti| = (t^2-1)^{1/2}$ onto itself
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When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ for $\Gamma$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least ...
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$C^{\infty}$ 1-form on a Riemann surface is unique.

Let $X$ be a Riemann surface and $\mathcal{A}$ be a complex atlas on $X$. Suppose that $C^{\infty}$ 1-forms are given for each chart of $\mathcal{A}$, which transform to each other on their common ...
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Algebraic curves and riemann surfaces

I am a physics undergrad with no formal background in complex analysis. I have done complex analysis at the level of the first 4 chapters (till Complex integration) from Churchill and Brown. I am very ...
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Why function $j(\tau)$ has degree 1?

We have $$j(\tau)=\frac{1}{q}+\sum_{n=0}^{\infty}a_nq^n, a_n\in\mathbb{Z},q=e^{2\pi i\tau}$$ Then it is said that because $j$'s only pole is simple, $j$ has degree 1 as a map ...
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Question about a property of elliptic function

Let $E=\mathbb{C}/\Lambda$ be a complex elliptic curve where $\Lambda=\omega_1\mathbb{Z}\oplus\omega_2\mathbb{Z}$ Let $f$ be a nonconstant elliptic function with respect to $\Lambda$. Let ...
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What does degree of an isogeny mean?

The book I'm reading doesn't provide the definition of degree of an isogeny and I failed to google it. Can anyone tell me?
Let $\Lambda$,$\Lambda'$ be two lattice in $\mathbb{C}$ and $m\neq 0\in\mathbb{C}$ satisfying $$m\Lambda\subset\Lambda'$$ The, the book I'm reading says that by the theory of finite Abelian groups ...