VALUE DISTRIBUTION THEORY RELATED TO NUMBER THEORY

By ISABELLA LEWIS

CONTENTS FIELD EXTENSIONS FIELDS WITH VALUATIONS ABSOLUTE VALUES EXTENSIONS OF ABSOLUTE VALUES DISCRIMINANT OF ?ELD EXTENSIONS DISCRIMINANT DEDEKIND DOMAIN DI?ERENT PRODUCT FORMULA HERMITIAN GEOMETRY GAUGES OF ELEMENTARY OPERATORS HYPERSURFACES NOCHKA WEIGHTS BASIC GEOMETRIC NOTIONS VARIETIES FUNCTION ?ELDS DIMENSIONS DI?ERENTIAL FORMS DIVISORS LINEAR SYSTEMS WEIL FUNCTIONS HEIGHTS IN NUMBER ?ELDS FUNCTORIAL PROPERTIES OF HEIGHTS 1 10 GAUSS’ LEMMA NOTIONS IN COMPLEX GEOMETRY HOLOMORPHIC FUNCTIONS COMPLEX MANIFOLDS MEROMORPHIC MAPPINGS HOLOMORPHIC VECTOR BUNDLES HOLOMORPHIC LINE BUNDLES HERMITIAN MANIFOLDS PARABOLIC MANIFOLDS INEQUALITIES ON SYMMETRIC POLYNOMIALS KOBAYASHI HYPERBOLICITY HYPERBOLICITY MEASURE HYPERBOLICITY OPEN PROBLEMS CHARACTERISTIC FUNCTIONS GROWTH OF RATIONAL FUNCTIONS LEMMA OF THE LOGARITHMIC DERIVATIVE SECOND MAIN THEOREM NOTES ON THE SECOND MAIN THEOREM THE CARTAN-NOCHKA THEOREM FIRST MAIN THEOREM FOR LINE BUNDLES 2 10 JACOBIAN SECTIONS 2 11 STOLL’S THEOREMS 2 12 CARLSON-GRI?THS-KING THEORY 2 12 1 SECOND MAIN THEOREM FOR LINE BUNDLES 2 12 2 GRI?THS’ AND LANG’S CONJECTURES ELLIPTIC CURVES THE GEOMETRY OF ELLIPTIC CURVES MODULAR FUNCTIONS CUSP FORMS PROBLEMS IN ELLIPTIC CURVES HYPERELLIPTIC AND RATIONAL CURVES THE ABC-CONJECTURE MORDELL’S CONJECTURE AND GENERALIZATIONS FERMAT EQUATIONS AND WARING’S PROBLEM THUE-SIEGEL-ROTH’S THEOREM SCHMIDT’S SUBSPACE THEOREM VOJTA’S CONJECTURES SUBSPACE THEOREMS ON HYPERSURFACES MAIN RESULTS AND PROBLEMS EQUIVALENCE OF THEOREMS 3 65 AND 3 68 PROOF OF THEOREM 3 65 PROOF OF THEOREM 2 63 VANISHING SUMS IN FUNCTION ?ELDS ALGEBRAIC FUNCTION ?ELDS MASON’S INEQUALITY NO VANISHING SUBSUMS NEVANLINNA’S THIRD MAIN THEOREM GENERALIZED MASON’S THEOREM GENERALIZED ABC-CONJECTURE GENERALIZED HALL’S CONJECTURE BOREL’S THEOREM AND ITS ANALOGUES BOREL’S THEOREM SIU-YEUNG’S THEOREM ANALOGUE OF BOREL’S THEOREM MEROMORPHIC SOLUTIONS OF FERMAT EQUATIONS WARING’S PROBLEM FOR MEROMORPHIC FUNCTIONS HOLOMORPHIC CURVES INTO A COMPLEX TORUS COMPLEX TORUS HYPERBOLIC SPACES OF LOWER DIMENSIONS PICARD’S THEOREM HYPERBOLIC CURVES HYPERBOLIC SURFACES UNIQUENESS POLYNOMIALS 4 10 FACTORIZATION OF FUNCTIONS 4 11 WIMAN-VALIRON THEORY EQUIDISTRIBUTION FORMULA SECOND MAIN THEOREM OF MEROMORPHIC FUNCTIONS EQUIDISTRIBUTION FORMULA FOR HYPERPLANES NON-ARCHIMEDEAN CARTAN-NOCHKA THEOREM HOLOMORPHIC CURVES INTO PROJECTIVE VARIETIES EQUIDISTRIBUTION FORMULA FOR HYPERSURFACES CHARACTERISTIC FUNCTIONS FOR DIVISORS THE ABC-THEOREM FOR MEROMORPHIC FUNCTIONS THE ABC-THEOREM FOR ENTIRE FUNCTIONS NON-ARCHIMEDEAN BOREL THEOREM WARING’S PROBLEM OVER FUNCTION ?ELDS 5 10 PICARD-BERKOVICH’S THEOREM VARIATIONS OF THE ?RST MAIN THEOREM GREEN’S FORMULA ANALOGUE OF VOJTA’S CONJECTURE MEROMORPHIC CONNECTIONS SIU THEORY SIU’S INEQUALITY GENERALIZATION OF SIU’S THEOREM BLOCH-GREEN’S CONJECTURE GREEN-GRI?THS’ CONJECTURE NOTES ON GRI?THS’ AND LANG’S CONJECTURES RIEMANN’S FUNCTIONAL EQUATION CONVERSE THEOREMS RIEMANN’S HYPOTHESIS HADAMARD’S FACTORIZATION NEVANLINNA’S FORMULA CARLEMAN’S FORMULA LEVIN’S FORMULA NOTES ON NEVANLINNA’S CONJECTURE