## Riemann surfacesThis text covers Riemann surface theory from elementary aspects to the fontiers of current research. Open and closed surfaces are treated with emphasis on the compact case. Basic tools are developed to describe the analytic, geometric, and algebraic properties of Riemann surfaces and the Abelian varities associated with these surfaces. Topics covered include existence of meromorphic functions, the Riemann -Roch theorem, Abel's theorem, the Jacobi inversion problem, Noether's theorem, and the Riemann vanishing theorem. A complete treatment of the uniformization of Riemann sufaces via Fuchsian groups, including branched coverings, is presented. Alternate proofs for the most important results are included, showing the diversity of approaches to the subject. For this new edition, the material has been brought up- to-date, and erros have been corrected. The book should be of interest not only to pure mathematicians, but also to physicists interested in string theory and related topics. |

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### Contents

CHAPTER II | 30 |

CHAPTER | 52 |

Hyperelliptic Riemann Surfaces | 93 |

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Abel's theorem abelian differentials algebraic analytic arbitrary assume automorphism base point canonical homology basis Choose closed curve compact Riemann surface compact support compact surface complex component compute conclude conformally equivalent consider coordinate vanishing Corollary cp(D defined denote dimension Dirichlet problem distinct points Div(M divisors of degree domain elliptic equation established exists finite fixed points Fuchsian Fuchsian group function elements genus g harmonic differentials harmonic function Hence holomorphic differentials holomorphic function holomorphic mapping hyperelliptic surface integral divisor isomorphic Lemma linear manifold meromorphic function metric Mobius transformation neighborhood non-constant non-gaps Note obtain ordP parabolic period matrix point of order polar divisor poles proj Proof Proposition recall Remark Riemann-Hurwitz Riemann-Roch theorem simply connected singularity subgroup subharmonic subset surface of genus theta function topological torus triangulation unique universal covering space vector Weierstrass points zero