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    National Tsing Hua University Institutional Repository > 理學院 > 數學系 > 博碩士論文  >  大域函數體的L函數的解析理論


    Please use this identifier to cite or link to this item: http://nthur.lib.nthu.edu.tw/dspace/handle/987654321/86600


    Title: 大域函數體的L函數的解析理論
    Authors: 莊治耘
    Chuang, Chih Yun
    Description: GH029521513
    博士
    數學系
    Date: 2014
    Keywords: 德林費爾德自守函數;大域函數體的虛二次擴張;大域函數體的維納-池原定理;志村曲線
    Drinfeld type automorphic forms;Imaginary quadratic fields over funciton fields;A Wiener-Ikehara Tauberian Theorem;Shimura curve
    Abstract: 論文的主要目的是建立Rankin-type的L-函數的functional equation對於奇特徵值的大域函數體。首先固定一個點叫做無窮遠點,我們考慮虛二次擴張對於這個無線遠點。在有理函數體中考慮某類特別德林費爾德的自守函數和某一種theta函數和某種character在ideal class group上的這種形式的funcitonal equation已經被Ruck and Tipp所證明。而這篇的工作是推廣到大域函數體。另一個部分是得到Wiener-Ikehara Tauberian theorem 對於大域函數體,這通常被用在解析統計上面的問題。我們應用此定理到漸進公式在divisor分布上。特別的是對於固定次方且每個質因式的次方都是偶次並且至多出現一次的多項式,我們有很好的漸進公式。
    The main part of this thesis aims at establishing the functional equa-
    tion of Rankin L-functions over arbitrary global function ?elds k with odd
    characteristic. Fixing an " in?nite" place ∞ of k, we consider an imaginary
    quadratic ?eld extension K/k (meaning ∞ does not split in K/k). Func-
    tional equation is proved for Rankin L-function formed by an automorphic
    cusp forms of Drinfeld type together with a theta function associated with
    given ideal class group character of K/k. This work generalizes previous
    results obtained by Ruck-Tipp (functional equation over the rational func-
    tion ?elds ). We also derive a Wiener-Ikehara Tauberian theorem for global
    function ?elds in the last chapter for use in the analytic problems concerning
    arithmetic statistics. Asymptotic formulas for counting positive divisors of
    a given function ?eld is investigated. This allows us to get in particular an
    asymptotic formula for the proportion of polynomials of a given degree over
    a ?nite ?eld which do not have odd degree irreducible factors.
    URI: http://nthur.lib.nthu.edu.tw/dspace/handle/987654321/86600
    Source: http://thesis.nthu.edu.tw/cgi-bin/gs/hugsweb.cgi?o=dnthucdr&i=sGH029521513.id
    Appears in Collections:[數學系] 博碩士論文

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