by Yan Cho Hung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 122-124). / Abstract also in Chinese. / Chapter 1 --- Complex algebraic curves --- p.6 / Chapter 1.1 --- Foundations --- p.6 / Chapter 1.1.1 --- Hilbert Nullstellensatz --- p.6 / Chapter 1.1.2 --- Complex algebraic curves in C2 --- p.9 / Chapter 1.1.3 --- Complex projective curves in P2 --- p.11 / Chapter 1.1.4 --- Affine and projective curves --- p.13 / Chapter 1.2 --- Algebraic properties of complex projective curves in P2 --- p.16 / Chapter 1.2.1 --- Intersection multiplicity --- p.16 / Chapter 1.2.2 --- Bezout's theorem and its applications --- p.18 / Chapter 1.2.3 --- Cubic curves --- p.21 / Chapter 1.3 --- Topological properties of complex projective curves in P2 --- p.23 / Chapter 1.4 --- Riemann surfaces --- p.26 / Chapter 1.4.1 --- Weierstrass &-function --- p.26 / Chapter 1.4.2 --- Riemann surfaces and examples --- p.27 / Chapter 1.5 --- Differentials on Riemann surfaces --- p.28 / Chapter 1.5.1 --- Holomorphic differentials --- p.28 / Chapter 1.5.2 --- Abel's Theorem for tori --- p.31 / Chapter 1.5.3 --- The Riemann-Roch theorem --- p.32 / Chapter 1.6 --- Singular curves --- p.36 / Chapter 1.6.1 --- Resolution of singularities --- p.37 / Chapter 1.6.2 --- The topology of singular curves --- p.45 / Chapter 2 --- Coding theory --- p.48 / Chapter 2.1 --- An introduction to codes --- p.48 / Chapter 2.1.1 --- Efficient noiseless coding --- p.51 / Chapter 2.1.2 --- The main coding theory problem --- p.56 / Chapter 2.2 --- Linear codes --- p.58 / Chapter 2.2.1 --- Syndrome decoding --- p.63 / Chapter 2.2.2 --- Equivalence of codes --- p.65 / Chapter 2.2.3 --- An introduction to cyclic codes --- p.67 / Chapter 2.3 --- Special linear codes --- p.71 / Chapter 2.3.1 --- Hamming codes --- p.71 / Chapter 2.3.2 --- Simplex codes --- p.72 / Chapter 2.3.3 --- Reed-Muller codes --- p.73 / Chapter 2.3.4 --- BCH codes --- p.75 / Chapter 2.4 --- Bounds on codes --- p.77 / Chapter 2.4.1 --- Spheres in Zn --- p.77 / Chapter 2.4.2 --- Perfect codes --- p.78 / Chapter 2.4.3 --- Famous numbers Ar (n，d) and the sphere-covering and sphere packing bounds --- p.79 / Chapter 2.4.4 --- The Singleton and Plotkin bounds --- p.81 / Chapter 2.4.5 --- The Gilbert-Varshamov bound --- p.83 / Chapter 3 --- Algebraic curves over finite fields and the Goppa codes --- p.85 / Chapter 3.1 --- Algebraic curves over finite fields --- p.85 / Chapter 3.1.1 --- Affine varieties --- p.85 / Chapter 3.1.2 --- Projective varieties --- p.37 / Chapter 3.1.3 --- Morphisms --- p.89 / Chapter 3.1.4 --- Rational maps --- p.91 / Chapter 3.1.5 --- Non-singular varieties --- p.92 / Chapter 3.1.6 --- Smooth models of algebraic curves --- p.93 / Chapter 3.2 --- Goppa codes --- p.96 / Chapter 3.2.1 --- Elementary Goppa codes --- p.96 / Chapter 3.2.2 --- The affine and projective lines --- p.98 / Chapter 3.2.3 --- Goppa codes on the projective line --- p.102 / Chapter 3.2.4 --- Differentials and divisors --- p.105 / Chapter 3.2.5 --- Algebraic geometric codes --- p.112 / Chapter 3.2.6 --- Codes with better rates than the Varshamov- Gilbert bound and calculation of parameters --- p.116 / Bibliography

Huang Yong Dong. / Thesis submitted in: November 2003. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 59-60). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 1.1 --- Abelian Varieties And Shimura Varieties --- p.5 / Chapter 1.2 --- Jacobians of Some Families of Curves --- p.7 / Chapter 1.3 --- Endomorphism Algebras of Jacobians of Curves --- p.8 / Chapter 2 --- Families of Abelian Varieties --- p.11 / Chapter 2.1 --- Abelian Varieties --- p.11 / Chapter 2.2 --- The Endomorphism Algebra of A Simple Abelian Varieties --- p.13 / Chapter 2.3 --- Family of Abelian Varieties and Shimura Varieties --- p.15 / Chapter 2.3.1 --- Real Multiplication --- p.16 / Chapter 2.3.2 --- Totally Indefinite Quaternion Multiplication --- p.19 / Chapter 2.3.3 --- Totally Definite Quaternion Multiplication --- p.22 / Chapter 2.3.4 --- Complex Multiplication --- p.25 / Chapter 2.3.5 --- Shimura Varieties --- p.28 / Chapter 2.4 --- The Endomorphism Algebra of A General Member --- p.29 / Chapter 3 --- Jacobians of Some Families of Curves --- p.32 / Chapter 3.1 --- Some Families of Curves --- p.32 / Chapter 3.2 --- Kodaira-Spencer Map --- p.37 / Chapter 3.3 --- Infinity of CM Type Points --- p.46 / Chapter 3.3.1 --- Φ Is Dominant --- p.46 / Chapter 3.3.2 --- Infinity of CM Points --- p.50 / Chapter 4 --- Endomorphism Algebras of Jacobians of Some Fam- ilies of Curves --- p.51 / Chapter 4.1 --- Jacobians Between Finite Coverings of Curves --- p.51 / Chapter 4.2 --- Endomorphism Algebras of Families of Jacobians --- p.53 / Chapter 4.2.1 --- The Case For μ = 1 --- p.53 / Chapter 4.2.2 --- The Case For μ = 2 --- p.55 / Chapter 4.2.3 --- The Case For μ =3 --- p.58 / Bibliography

Lau, Sui Ki. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 81). / Abstract also in Chinese. / Chapter 1 --- Basic Notions of Riemann Surfaces --- p.5 / Chapter 1.1 --- "Functions, Forms and Hurwitz's Formula" --- p.6 / Chapter 1.2 --- Divisors --- p.10 / Chapter 1.3 --- Plucker's Formula for a Smooth Projective Plane Curve --- p.14 / Chapter 1.4 --- Sheaves and Cohomology --- p.17 / Chapter 2 --- The Riemann-Roch Theorem and Algebraic Curves --- p.27 / Chapter 2.1 --- Finiteness Theorem --- p.27 / Chapter 2.2 --- Transcendence Degree of M(X) --- p.33 / Chapter 2.3 --- The Riemann-Roch Theorem and Serre Duality --- p.37 / Chapter 2.4 --- Holomorphic Embedding in a Projective Space --- p.44 / Chapter 2.5 --- Algebraic Curves --- p.50 / Chapter 3 --- Invertible Sheaves and Line Bundles --- p.55 / Chapter 3.1 --- Algebraic Sheaves --- p.55 / Chapter 3.2 --- Invertible Sheaves --- p.55 / Chapter 3.3 --- Line Bundles --- p.61 / Chapter 3.4 --- Isomorphic Representations of the Picard Group --- p.66 / Chapter 4 --- A Uniqueness Theorem for Algebraic Curves --- p.72 / Chapter 4.1 --- Associated Curves and Normal Forms --- p.72 / Chapter 4.2 --- Proof of a Uniqueness Theorem for Algebraic Curves --- p.76 / Bibliography --- p.81