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By Ernesto Girondo

Few books with regards to Riemann surfaces conceal the really smooth thought of dessins d'enfants (children's drawings), which used to be introduced via Grothendieck within the Nineteen Eighties and is now an energetic box of analysis. during this e-book, the authors start with an undemanding account of the idea of compact Riemann surfaces seen as algebraic curves and as quotients of the hyperbolic airplane by way of the motion of Fuchsian teams of finite style. They then use this data to introduce the reader to the speculation of dessins d'enfants and its reference to algebraic curves outlined over quantity fields. loads of labored examples are supplied to assist realizing, so no adventure past the undergraduate point is needed. Readers with none past wisdom of the sphere of dessins d'enfants are taken speedily to the leading edge of present learn.

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A2g+1 , 0), ∞. The branching order at these points equals 2, and the corresponding branch values are a1 , . . , a2g+1 , ∞. As a meromorphic function, x has a single pole of multiplicity 2 at ∞. Its zeros are located at the points Q± = 0, ± (−ai ) , and both are simple. As for the function y, similar computations show that all the zeros are simple and located at the 2g+1 points (a1 , 0), . . 1 Basic definitions 19 whereas ∞ is the unique pole (of multiplicity 2g + 1). The order of the zeros and poles follow from the computations ⎛ ⎞ ⎝ 2 y ◦ ϕ−1 i (z) = y z + ai , z z 2 + ai − ak ⎠ k=i z 2 + ai − ak = z k=i and ⎛ y ◦ ψ −1 (z) = y ⎝ = 2g+1 1 1 , z 2 z 2g+1 ⎞ 1 − ak z 2 ⎠ k=1 2g+1 1 1 − ak z 2 z 2g+1 k=1 respectively.

2 Topology of Riemann surfaces 45 called the degree of f , is independent of the choice of c ∈ C. Proof Let c = ∞ and P ∈ f −1 (c) a point of branching order mP (f ) = n. Then the Taylor expansion of f − c around P is n n+1 (f − c) ◦ ϕ−1 + ··· j (z) = an z + an+1 z hence (f − c) ◦ ϕ−1 j (z) ◦ ϕ−1 j (z) (f − c) = n + ··· z d(f − c) with (f − c) residue n. Similarly, when P is a pole of f , the residue of ω at P equals −mP (f ). 31). If we take multiplicities into account this number remains constant at all values c ∈ P1 .

For that we need to say something about the integration of differential forms. If c : I = [0, 1] → S is a (piecewise differentiable) path and ω is a meromorphic differential without poles in c(I), we can compute the integral c ω. The result does not depend on the choice of coordinates, since if γ = c|[a,b] is a small piece of c lying inside Ui ∩ Uj the two possible ways to compute γ ω, namely ω= ϕi ◦γ γ fi ◦ ϕ−1 i and ω= ϕj ◦γ γ fj ◦ ϕ−1 j give the same result. 55 Let P be a a point lying in a chart (Uj , ϕj ).

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