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。 ;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.
