For questions about Riemann surfaces, that is compact analytic manifolds of (complex) dimension 1, and related topics.

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Harmonic maps between Riemann surfaces

In 'Compact Riemann surfaces' Jost defines harmonic maps between surfaces $S_1,S_2$, with local coordinates z on $S_1$ and metric $\rho^2|du\,d\overline{u}|$ on $S_2$ as $u\in C^2$ solving the ...
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Use differential form to prove meromorphic function on compact riemann surface has same zeros and poles

I am reading mine's modular form note, proposition 1.12 states that the sum of residues of a differential form on compact Riemman surface is 0. Then he states that applies this to $df/f$, then we can ...
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1answer
54 views

Using the Maximum Modulus Principle to prove that every holomorphic function on a compact Riemann surface is constant

I have read in a number of sources (including here) that a holomorphic function on a compact Riemann surface must be constant. The reason given has always been the Maximum Modulus principle, but ...
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Complex Atlas for Elliptic Curves over $\mathbb{C}$

I know that every elliptic curve over $\mathbb{C}$ is isomophic to a torus $\mathbb{C}/\Lambda$ in the sense of Riemann Surfaces, moreover $E(\Lambda)$ as topological subspace of ...
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2answers
98 views

On the Usual Orientation of Cubic Graphs in Random Construction of Riemann Surfaces

In "Random Construction of Riemann Surfaces", Robert Brooks and Eran Makover say : Definition 2.1 A left-hand turn path on $(\Gamma, \mathcal O)$ is a closed path on [the cubic graph] $\Gamma$ ...
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Fundamental domain of a Fuchsian group is non-compact if the group contains parabolic element

Suppose a discrete subgroup $\Gamma$ of $PSL(2,\mathbb{R})$ acts on $\mathbb{H}^2$. Why is the fundamental domain non-compact if $\Gamma$ contains a parabolic element? Thanks in advance.
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1answer
40 views

Choice of Fundamental Domain of Torus (Dehn Twists?)

So I would like to consider a lattice $\Lambda \subseteq \mathbb{C}$ generated by $(1,\tau)$ with $\tau$ in the upper-half complex plane. This lattice $\Lambda$ will remain fixed. If you choose the ...
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Construct the Riemann surface of the function $f(z)=((z-1)(z-2)(z-3))^{2/3}$.

Construct the Riemann surface of the function $f(z)=((z-1)(z-2)(z-3))^{2/3}$. 1,2 and 3 are branch points. But how can we determine $\infty$ is also a branch point. And how can we determine branch ...
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Subgroups of $\text{PSL}(2, \mathbb{R})$ Closed under Transposition

I am wondering, does anyone know if there is a classification of transposition-closed (Fuchsian) subgroups of $\text{PSL}(2, \mathbb{R})$? I can't read French, so for all I know it's sitting in the ...
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1answer
54 views

Divisors of degree $2g-2$ on a hyperelliptic curve of genus $g$

Suppose I have a divisor $D$ of degree $2g-2$ on a hyperelliptic curve of genus $g$. Then I can prove that either a) $K_C\otimes\mathcal{O}(-D)=\mathcal{O}_C$, that is $K_C\cong \mathcal{O}(D)$, or ...
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How do I get $ \int_0^1 \frac{dz}{\sqrt{z(z - 1\,)(z+1\,)}} = \frac{\sqrt{\pi}}{2} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{9}{4})}$?

While reading physics papers I found a very interesting integral so I decided to write it down. Let $p(z) = z^ 3 - 3\Lambda^ 2 z$ where $\Lambda$ could be any number. If you want $\Lambda = 1$ and ...
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Generators of commutator subgroup of fundamental group of genus-2 surface

Recall the fundamental group of a genus-2 surface: $$ \pi_1(\Sigma_2) = < a_1, b_1, a_2, b_2 \mid [a_1, b_1][a_2, b_2] = 1 > $$ By which I mean a free group of four variables, divided out by ...
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1answer
66 views

The lattice of $y^2=x(x-1)(x-λ)$

We know that every elliptic curve is associated with a lattice. So is the lattice of $y^2=x(x-1)(x-λ)$ just the lattice spanned by $\{0,1,λ\}$? If yes, is there some direct explanation? (Do not ...
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51 views

Is the function $f(z)=z^2$ injective?

I know this may sound silly, but I'm not so sure if the method of constructing Riemann surfaces for the domain also applies to the codomain. For example, we can make the function $f(z)=\sqrt(z)$ ...
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1answer
18 views

Degree one branched cover is a homeomorphism

Suppose that $f:X \to Y$ is a branched cover of Riemann surfaces and a covering map of degree one outside of the ramification points. Then is $f$ a homeomorphism?
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Producing Abelian differentials with order $0$ at specified points.

On a compact Riemann surface, I wish to find an Abelian differential (meromorphic $1$-form) which does not have zeroes or poles (order $0$) at a finite number of pre-arranged points. This is stated ...
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1answer
35 views

Doubt with an illustration of algebraic curves and Riemann surfaces

The complex equation $w - z = 0$, $z$, $w \in \mathbb{C}$, represents a complex curve (also called $1$-dimensional complex manifold). This complex curve corresponds to the complex plane $\mathbb{C}$ ...
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33 views

Riemann surface for $f(z)=\sqrt{z^2+1}$

How would I determine the form of the Riemann surface for the function $f(z)=\sqrt{z^2+1}$ taking the branch cut to be the imaginary axis for $Im(z)\in[-i,i]$? And is there anyway to plot this using ...
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What is a cusp neighborhood corresponding to a parabolic Mobius transformation in a Riemann surface?

I am referring to this wikipedia entry. So what I understand is that they are defining it using the Fuchsian model. If $\Gamma$ is a Fuchsian group, its parabolic elements correspond to the cusps of ...
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First non-zero eigenvalue of the Laplacian

Consider the Laplacian $-\Delta$ on a compact Riemann surface of genus $g \geq 2$. Is there an upper bound on the multiplicity of the first non-zero eigenvalue of the Laplacian? In particular, does it ...
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1answer
78 views

Is every Riemann surface a 2-sheeted covering?

Given an algebraic curve $X$ over $\mathbb{C}$, i.e. a Riemann surface and a fixed set of pairs of points $S=\{(p_1,q_1),...,(p_1,q_1)\}$ is there an algebraic curve Y, possibly singular, and a map ...
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1answer
27 views

Bounds on area of geodesic disk

Are there upper and lower bounds, in terms of curvature, on the area of geodesic disks on compact smooth 2-manifolds embedded (or immersed) in $\mathbb{R}^3$? It is tempting to imagine bounds must ...
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Why the modular curve $X(1)$ has genus 0?

I'm reading "A first course in modular forms" by Diamond and it is mentioned that the modular curve $X(1)=\mathrm{SL}_{2}(\mathbb{Z})\backslash\mathcal{H}^{*}$ has genus 0 (here $\mathcal{H}^{*}$ is ...
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1answer
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Gluing together Riemann surfaces, don't see why $Z$ is union of two compact sets.

Consider Lemma 1.7 from page 60-61 of Miranda's Algebraic Curves and Riemann Surfaces. For a link to the book, see here. Lemma 1.7. With the above construction, $Z$ is a compact surface of genus ...
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1answer
12 views

Transition of complex charts are biholomorphic [Riemann Surfaces, Miranda Exercise]

I'm reading Rick Miranda, Algebraic Curves and Riemann Surfaces. I'm having problems with this exercise about complex charts. Let $\phi_i: U_i \to V_i$, $i=1,2$, be complex charts on $X$ with ...
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1answer
33 views

simple question about the degree of a divisor on riemann surface

Let $C=H \cap S$ where $S$ is a surface of general type canonically embedded in $\mathbb{P}^N$ for some $N>0$ and $H$ the divisor of an hyperplane in $\mathbb{P}^N$. $H_{|C}$ is a divisor on $C$. ...
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37 views

Riemann - Hurwitz Formula for topology.

I am quite confused about the notion of branch points at infinity? and even in general the idea of branch points? I know branch points to be where points diverges to infinity. Could someone please ...
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29 views

Understanding how a global analytic function is continuous on its Riemann surface (Ahlfors)

In Ahlfors Complex Analysis on page 278 he explains when we can say that the function (global analytic) is continuous on $\Gamma$ (it's Riemann surface). I'm pretty new to this and can't follow his ...
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1answer
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How do I prove that $\int_0^1 \frac{1}{(x^2-x^3)^{1/3}} =\frac{2\pi}{\sqrt{3}}$?

This is a problem from Mathematical Methods for Physicists, by Arfken, 7th edition (Problem 11.8.27). I know the integrals in the circular paths around 0 and 1 will vanish, but am completely lost on ...
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1answer
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An $d$-unramified covering of compact Riemann surfaces induce a (monodromy) action on $d$ letters. Is the opposite true?

Let $S_1, S$ be compact connected Riemann surfaces, $f : S_1 \rightarrow S$ be a meromorphic function of degree $d$ that branch over $B \subset S$. The unmarried covering $f : S_1 \backslash f^{-1}(B) ...
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Theta characteristic on curves of genus zero and one

The aim of that question is how to compute theta characteristic (i.e a line $L$ bundle such that $L^{\otimes2}=\omega_C$ where $\omega_C$ is the canonical divisor of my surface) on complex compact ...
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CFT's vs Vertex Operator Alagebras

I am trying to clear my ideas about the relation between a Conformal Field Theory (CFT) and a Vertex Operator Algebra (VOA). For me a CFT based on a (complex) vector space $H$ is a projective monoidal ...
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Differential 1-form on Riemann surfaces

Let $X$ be a smooth affine plane curve defined by $f(u,v)=0$. Show that $(\partial{f}/\partial{u})du=-(\partial{f}/\partial{v})dv$ as holomorphic 1-forms on $X$. I tried solving this question using ...
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1answer
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Condition for ramification points of a projection $\pi : X \to \mathbb{P}^1 $

Let $X = \{F=0\} \subset \mathbb{P}^2$ be a projective plane curve and let $\pi : X \to \mathbb{P}^1$ be defined by $\pi [x:y:z] \to [x:y]$. I'm trying to understand why the following is true: ...
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1answer
25 views

What are the morphisms in the category of unramified coverings over a compact Riemann surface?

Fix a compact Riemann surface $S$, and finite a set of branch points $B \subseteq S$. Consider the collection of Riemann surfaces $S_1$ and mermorphic functions $f: S_1 \rightarrow S$, such that $f$ ...
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3answers
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Is the fundamental group of a compact Riemann surface *after* removing a finite number of points still a Fuchsian group?

Let $S$ be a compact R.S. admitting a Fuchsian model $\mathbb{H} / \Gamma$. We know that $\pi_1(S) \cong \Gamma$. Let $\mathcal{B} \subseteq S$ be a finite set of points, is $\pi_1(S - \mathcal{B})$ ...
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Establishing Linear Equivalence of Divisors on Curves

I am trying to do some questions from Geometry of Algebraic Curves by Arbarello-Cornalba-Griffiths-Harris. Here are some of the examples: Exercise A3: Curve: $y^2=x^3+1$. Let $\Gamma=C$ be the ...
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Riemann surface with $w=z-\sqrt{z^2-1}$ single-valued

I'm trying to find the Riemann surface that makes $w=z-\sqrt{z^2-1}$ single-valued, but I'm not sure how to approach the problem.
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1answer
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Under what conditions do there exist non-constant meromorphic functions between general Riemann surfaces?

The uniformization theorem answers this question for particular Riemann surfaces, but do we have a general theorem for this? Do we also get meromorphic functions that can separate points?
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Meromorphic function written as Weierstrass Elliptic Function [closed]

Let $\Lambda$ be a lattice in the complex plane. And Weierstrass Elliptic Function $$\wp(z)=\frac{1}{z^2}+\sum_{\omega \in \Lambda - \{0\}}\frac{1}{(z-\omega)^2}-\frac{1}{\omega ^2}$$ How can I ...
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1answer
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Are there closed Riemann surfaces without non-constant holomorphic functions?

I came across the Handbook of Teichmuller Theory, and they talk about "closed Riemann surfaces with non-constant holomorphic functions". Are there Riemann surfaces without those functions?
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Show $f$ is an immersion, where $f$ holomorphic map between compact Riemann Surfaces

I am trying to show that: If $f: X \to Y$ is a non-constant holomorphic map between compact Riemann surfaces, of degree $1$, then $f$ is an immersion. I tried proving this by contradiction. If ...
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cohomology of a tangent bundle

Suppose that $C$ is a complex riemann surface of positive genus lying in a complex algebraic surface of general type. Let $T_C$ the tangent bundle to the curve $C$. Is there a way to compute the ...
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1answer
286 views

Is there a complex surface into which every Riemann surface embeds?

Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma \to \mathbb{CP}^3$. It follows from the ...
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1answer
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Deduce riemannian mapping theorem for domains of $\hat{\mathbb{C}}$ from uniformization theorem

I want to use the uniformization theorem, which states that every simply connected riemann surface is biholomorphically equivalent to either the riemann sphere $\hat{\mathbb{C}} = \mathbb{C} \cup ...
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1answer
30 views

Bijective conformal maps from a torus to itself

I would think of torus as follows: in the complex plane $\mathbb{C}$, consider two $\mathbb{R}$-independent vectors $\{v_1,v_2\}$. Then $v_1,v_2$ together with $0$ will determine a parallelogram, and ...
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1answer
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What does the compatibility condition in the definition of meromorphic differentials mean?

Let $S$ be a Riemann surface, with an atlas $(U_i, \varphi_i)_{i \in I}$. For any $P \in S$, denote $$\frac{dz_i}{dz_j}(P):= (\varphi_i \circ \varphi_j^{-1})^\prime (\varphi_j(P)).$$ We then define a ...
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Can the Klein Quartic be parameterized by meromorphic upper-half plane functions?

It is known that Elliptic Curves in canonical form can be parameterized by the Weierstrass elliptic function and its derivative on a suitably chosen lattice: $$[\wp'(z)]^2 = ...
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tangent function determines a local homeomorphism

The question ask me to show that tangent function determines a local homeomorphism $\tan: \mathbb C \to \mathbb C P^1$. I don't understand what the question asking, is the question asking me to show ...
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61 views

Understanding Complex Differentials (forms)

In the study of Riemann surfaces, many books bring in their discussions, the complex differentials or differential forms, and there my understanding gets stopped. I personally interacted with many ...