## Classical Theory of Compact Riemann Surfaces |

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a-place Abel's theorem abelian abelian group Algebraic functions apply assertion avoids the poles canonical divisor complex analytic complex structure conformal equivalence consider the mapping constants COROLLARY cycles defined deg f deg ox denote differential of second dim ox dimension g dimensional Dirichlet principle disc dz/z elliptic entials exists finite number fixed point follows formula hence dim Hodge theorem holomorphic differential holomorphic quadratic differentials integral INVERSION THEOREM kernel LEMMA linear meromorphic differential neighborhood non-constant function non-constant meromorphic function non-zero function f normalized differential number of poles one-form one-one period relations pole of order Proof proved rational function real harmonic residue RESIDUE THEOREM resp result Riemann surface Riemann-Roch theorem satisfying second kind Section simple poles space of holomorphic subspace Suppose theory topology torus triangulation uniform!zing uniformizing variable Vp f whole number write zeros and poles