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研究生:魏福村
研究生(外文):Wei, Fu-Tsun
論文名稱:OnArithmeticofCurvesoverFunctionFields
指導教授:蔡孟傑于靖于靖引用關係
指導教授(外文):Meng Kiat ChuahYu, Jing
學位類別:博士
校院名稱:國立清華大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
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 fieldquaternion algebraautomorphic formelliptic 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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