## Algebraic functions: 1964 lectures delivered at Columbia University |

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1-simplex contained Abel's theorem Abelian Integrals algebraic closure ALGEBRAIC FUNCTIONS analytic manifold Bilinear bounded length modulo coefficient Columbia University Serge complex numbers constant curve cycles define the integral define z(P denote differential discrete valuation ring divisor of degree divisors of functions dz/z equal factor space Find a dfk finite number g distinct points g-dimensional space genus group of divisors Hence holomorphic function induces infinity inside the polygon Jacobian kernel LEMMA linear relation linearly maximal ideal meromorphic function Mg is non-special modulo periods number of primes number of sheets number of valuation open set orientation parameter at Q path point a,b polynomial power series Proof prove Jacobi's theorem pure imaginary ramification index ramified vertex Q resp(F<j Riemann surface Riemann-Roch theorem small neighborhood suffices to prove sum is taken Surface as Analytic surjectivity t-plane Theorem 9 thereby proving triangles uniformization theorem shows