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研究生： 魏福村
Wei, Fu-Tsun
論文名稱： On Arithmetic of Curves over Function Fields
指導教授： 蔡孟傑
Meng Kiat Chuah
于靖
Yu, Jing
口試委員:
學位類別： 博士
Doctor
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2010
畢業學年度： 99
語文別： 英文
論文頁數： 114
中文關鍵詞： 函數體 、四元數代數 、自守型 、橢圓曲線
外文關鍵詞： function field, quaternion algebra, automorphic form, elliptic curves
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    • There are two parts in the thesis.
    Part One (Chapter 1 to 4) is on arithmetic of definite Shimura curves over function fields and automorphic forms.
    We construct certain theta series from definite quaternion algebras over function fields which generate
    the space of harmonic automorphic forms.
    From the special points on definite Shimura curves and those theta series
    we deduce the critical central value of $L$-series of automoprhic forms.
    As applications, an analogue of Waldspurger's formula and
    critical central values of Hasse-Weil $L$-function of certain elliptic curves over function fields are obtained.

    Part Two (Chapter 5) is a published paper: On the independence of Heegner points over function fields.
    We prove the independence of Heegner points for different "imaginary" quadratic function fields
    and get a subgroup of elliptic curves with arbitrary large rank.

    Acknowledgements Introduction Part 1. Definite Shimura Curves Chapter I. Preliminaries 1. Drinfeld modules and isogenies 2. Finite Drinfeld modules 2.1 Supersingular Drinfeld modules 2.2 Mass formula Chapter II Brandt Matrices and Definite Shimura Curves 1. Brandt matrices 1.1 Trace formula 1.2 Supersingular Drinfeld modules and Brandt matrices 1.3 Recurrence relations of Brnadt matrices 1.4 Theta series 2. Definite Shimura curves 3. Actions on Gross points 4. Hecke correspondence and Gross height pairing Chapter III Automorphic Forms of Drinfeld Type and L-Series 1. Automorphic forms of Drinfeld type and main theorem 1.1 Automorphic forms of Drinfeld type 1.2 Hecke operators 1.3 Main theorem 2. J-L correspondence and the multiplicity one theorem 2.1 Jacquet-Langlands correspondence 2.2 Multiplicity one theorem 3. Special values of L-series 3.1 Rankin's method 3.2 The heights of special points 3.3 The special value \Lambda(f, \chi,0) Chapter IV. Integral Weight and Half Integral Weight 1. Integral weight 1.1 Operator T_{\infty, \kappa} 1.2 Automorphic forms of weight 2 2. Half integral weight 2.1 Theta series 2.2 Extension G of GL_2(k_{\infty}) by S^1 2.3 Half integral weight and operators T_{\infty, kappa/2} 2.4 Hecke operators 2.5 A three squares problem 2.6 An analogue of Waldspurger's formula Part 2. Elliptic Curves and Heegner Points Chapter V. On the independence of Heegner points over function fields 1. Drinfeld modular curves 1.1 Analytic theory of Drinfeld modules 1.2 Moduli spaces and Drinfeld modular curves 1.3 CM-points associated to O_K 2. Independence of Heegner points 2.1 Independence property 2.2 Proof of Claim I 2.3 Proof of Claim II 3. Existence of Large Prime-to-2p Part of Class Number 3.1 Odd characteristic cases 3.2 Even characteristic cases 3.3 Asymptotic behavior Bibliography Symbols in Part I

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