博碩士論文 942201001 詳細資訊




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姓名 洪斌哲(Pin-Chi Hung)  查詢紙本館藏   畢業系所 數學系
論文名稱 一些關於L函數中心值不為零的結果
(Some results on the non-vanishing of central L-values)
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摘要(中) 尋找代數簇在數域中的有理解是數論中最古老的問題之一。著名的斯溫納頓-戴爾猜想預測了代數簇有理解的存在性和他們所連繫的L函數特殊值
等不等於零有關連。因此,了解L函數的特殊值是不是為零一直是讓人感興趣的。在這篇博士論文裡面,我們證明了一些CM橢圓曲線以及CM體括張上面
希爾伯特模型式的中心值不為零。

  在論文的第一部份,我們證明了某些CM橢圓曲線上面的有理解個數是有限的藉由證明他們的漢斯-魏爾L函數的中心值不為零。藉由表現理論,我們
證明了這些中心值不為零充分必要於某些自守模型式的彼得松範數不為零。因此,我們利用這些彼得松範數的解析估計去證明第一個主要結果。

  我們第二個結果是關於對割圓扭變的希爾伯特模型式的中心值模一個質數l不為零的結果。這個結果可以應用來證明某種蘭金-塞爾伯格摺積岩澤主猜想,
故是史金納用來證明柯立瓦根以及格羅斯-察吉爾反定理的工具之一。利用瓦爾斯皮傑公式,我們證明了L函數中心值不為零等價於一個在定四元數代數上的模型式
對於CM點的加權和不為零。因此,我們利用科尼爾-瓦斯塔爾關於零維度志村簇上CM點的均勻分布的工作來證明我們的第二個結果。
摘要(英) One of the oldest problems in number theory is to find rational points of algebraic varieties over number fields.
The famous Birch and Swinnerton-Dyer conjecture predict that the existence of rational points of algebraic varieties
is closely related to the vanishing/non-vanishing of special values of the associated L-functions. Therefore, it is
always interesting to know whether special values of L-functions are non-vanishing. In this thesis, we investigate
the non-vanishing of central L-values for certain CM elliptic curves and Hilbert modular forms over CM fields.

In the first part, we prove the finiteness of rational points of some CM elliptic curves by showing the
non-vanishing of the central L-values of their Hasse-Weil L-functions. Using representation theory,
we prove that the central L-values are non-zero if and only if the Petersson norm of some automorphic forms are non-zero.
Therefore, our first main result follows from the analytic estimate of these Petersson norms.

Our second result is on the non-vanishing modulo l of central L-values with anticyclotomic twists for Hilbert modular forms.
This result will have application to Iwasawa main conjecture for certain Rankin-Selberg convolution which serves a key ingredient
in Skinner′s proof on the converse of Kolyvagin and Gross-Zagier. By Waldspurger′s formula, the non-vanishing of the central L-value
is equivalent to a weighted sum of a newform on some definite quaternion algebra over CM points. Then, our second main result follows
from using the work of Cornut-Vatsal
on the uniform distribution of CM point in zero dimensional Shimura varieties.
關鍵字(中) ★ 橢圓曲線
★ 模型式
★ L函數
關鍵字(英) ★ elliptic curve
★ modular form
★ L-function
論文目次 目錄
中文摘要..................................................I
Abstract................................................III
謝誌......................................................V
目錄....................................................VII
Chapter 1. Introduction................................... 1
1. Canonical CM elliptic curves with quadratic twists…. 1
2. Hilbert modular forms with anticyclotomic twists..... 3
Chapter 2. Central L-values of canonical CM elliptic curves
with quadratic twists.................................... 7
1. Notation and definitions............................ 7
2. Explicit Rallis inner product formula................ 8
3. Eigenfunctions of Weil representations of U(1) over
dyadic fields...................................... 12
4. Explicit formulae of central Hecke L-values in terms of
theta functions.................................... 26
5. Nonvanishing of central L-value of canonical characters
with quadratic twists............................... 28
Chapter 3. Central L-values with anticyclotomic twists for
Hilbert modular forms.................................... 45
1. Notation and definitions........................... 45
2. Gross points and modular forms on definite quaternion
algebras........................................... 47
3. Special value formula.............................. 50
4. Theta elements and p-adic L-functions................ 72
5. The non-vanishing of theta elements modulo l......... 77
參考文獻................................................ 89
參考文獻 中文摘要..................................................I
Abstract................................................III
謝誌......................................................V
目錄....................................................VII
Chapter 1. Introduction................................... 1
1. Canonical CM elliptic curves with quadratic twists…. 1
2. Hilbert modular forms with anticyclotomic twists..... 3
Chapter 2. Central L-values of canonical CM elliptic curves
with quadratic twists.................................... 7
1. Notation and definitions............................ 7
2. Explicit Rallis inner product formula................ 8
3. Eigenfunctions of Weil representations of U(1) over
dyadic fields...................................... 12
4. Explicit formulae of central Hecke L-values in terms of
theta functions.................................... 26
5. Nonvanishing of central L-value of canonical characters
with quadratic twists............................... 28
Chapter 3. Central L-values with anticyclotomic twists for
VIII
Hilbert modular forms.................................... 45
1. Notation and definitions........................... 45
2. Gross points and modular forms on definite quaternion
algebras........................................... 47
3. Special value formula.............................. 50
4. Theta elements and p-adic L-functions................ 72
5. The non-vanishing of theta elements modulo l......... 77
參考文獻................................................ 89
指導教授 謝銘倫(Ming-Lun Hsieh) 審核日期 2014-6-3
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