published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 1998. / Includes bibliographical references (leaves 110-112).

This thesis is comprised of three logically independent parts. As the title suggests, each part is related to vector bundles on abelian varieties. We first use Brill-Noether theory to study the geometry of a general curve in its canonical embedding. We prove that there is no $g$ for which the canonical embedding of a general curve of genus $g$ lies on the Segre embedding of any product of three or more projective spaces. We then consider non-degenerate line bundles on abelian varieties. Central to our work is Mumford's index theorem. We give an interpretation of this theorem, and then prove that non-degenerate line bundles, with nonzero index, exhibit positivity analogous to ample line bundles. As an application, we determine the asymptotic behaviour of families of cup-product maps. Using this result, we prove that vector bundles, which are associated to these families, are asymptotically globally generated. To illustrate our results, we consider explicit examples. We also prove that simple abelian varieties, for which our results apply in all possible instances, exist. This is achieved by considering a particular class of abelian varieties with real multiplication. The final part of this thesis concerns the theory of theta and adelic theta groups. We extend and refine work of Mumford, Umemura, and Mukai. For example, we determine the structure and representation theory of theta groups associated to a class of vector bundles which we call simple semi-homogeneous vector bundles of separable type. We also construct, and clarify functorial properties enjoyed by, adelic theta groups associated to line bundles. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2013-06-24 17:14:21.687

Yu, Hok Pun. / Thesis submitted in: October 2008. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 55-56). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Complex Tori --- p.8 / Chapter 2.1 --- Homomorphisms of complex tori --- p.9 / Chapter 2.2 --- Cohomology of Complex Tori --- p.10 / Chapter 3 --- Line bundles on complex tori --- p.11 / Chapter 3.1 --- First Chern classes --- p.11 / Chapter 3.2 --- Semicharacters on line bundles --- p.12 / Chapter 3.3 --- Theorem of the Square --- p.14 / Chapter 4 --- Principally polarized abelian varieties --- p.16 / Chapter 4.1 --- Riemann Relations --- p.17 / Chapter 4.2 --- Characteristics of line bundles --- p.20 / Chapter 4.3 --- Theta Functions --- p.21 / Chapter 4.4 --- The Ox(l) bundle --- p.22 / Chapter 4.5 --- Metric on Ox(l) --- p.23 / Chapter 4.6 --- Abelian Varieties and Elliptic Curves --- p.24 / Chapter 5 --- Isogeny of Abelian Varieties --- p.26 / Chapter 5.1 --- Symmetric Line Bundles --- p.27 / Chapter 5.2 --- Theta Relations --- p.28 / Chapter 5.3 --- Theta Divisors --- p.30 / Chapter 6 --- Jacobians --- p.32 / Chapter 6.1 --- Jacobian as an abelian variety --- p.33 / Chapter 6.2 --- Abel-Jacobi Theorem --- p.36 / Chapter 6.3 --- Torelli´ةs theorem --- p.42 / Chapter 7 --- The Heisenberg Group --- p.43 / Chapter 8 --- Balanced Embedding into the Projective Space --- p.50

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published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

Doctor of Philosophy / The Hecke character of an abelian variety A/F is an isogeny invariant and the Galois action is such that A is isogenous to its Galois conjugate A^σ if and only if the corresponding Hecke character is fixed by σ. The quadratic twist of A by an extension L/F corresponds to multiplication of the associated Hecke characters. This leads us to investigate the Galois groups of families of quadratic extensions L/F with restricted ramification which are normal over a given subfield k of F. Our most detailed results are given for the case where k is the field of rational numbers and F is a field of definition for an elliptic curve with complex multiplication by K. In this case the groups which occur as Gal(L/K) are closely related to the 4-torsion of the class group of K. We analyze the structure of the local unit groups of quadratic fields to find conditions for the existence of curves with good reduction everywhere. After discussing the question of finding models for curves of a given Hecke character, we use twists by 3-torsion points to give an algorithm for constructing models of curves with known Hecke character and good reduction outside 3. The endomorphism algebra of the Weil restriction of an abelian variety A may be determined from the Grössencharacter of A. We describe the computation of these algebras and give examples in which A has dimension 1 or 2 and its Weil restriction has simple abelian subvarieties of dimension ranging between 2 and 24.

Doctor of Philosophy / The Hecke character of an abelian variety A/F is an isogeny invariant and the Galois action is such that A is isogenous to its Galois conjugate A^σ if and only if the corresponding Hecke character is fixed by σ. The quadratic twist of A by an extension L/F corresponds to multiplication of the associated Hecke characters. This leads us to investigate the Galois groups of families of quadratic extensions L/F with restricted ramification which are normal over a given subfield k of F. Our most detailed results are given for the case where k is the field of rational numbers and F is a field of definition for an elliptic curve with complex multiplication by K. In this case the groups which occur as Gal(L/K) are closely related to the 4-torsion of the class group of K. We analyze the structure of the local unit groups of quadratic fields to find conditions for the existence of curves with good reduction everywhere. After discussing the question of finding models for curves of a given Hecke character, we use twists by 3-torsion points to give an algorithm for constructing models of curves with known Hecke character and good reduction outside 3. The endomorphism algebra of the Weil restriction of an abelian variety A may be determined from the Grössencharacter of A. We describe the computation of these algebras and give examples in which A has dimension 1 or 2 and its Weil restriction has simple abelian subvarieties of dimension ranging between 2 and 24.