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    Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/64569

    Title: 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
    Authors: 洪斌哲;Hung,Pin-Chi
    Contributors: 數學系
    Keywords: 橢圓曲線;模型式;L函數;elliptic curve;modular form;L-function
    Date: 2014-06-03
    Issue Date: 2014-08-11 18:43:03 (UTC+8)
    Publisher: 國立中央大學
    Abstract: 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


    ;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.
    Appears in Collections:[數學研究所] 博碩士論文

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