摘要
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。
並列摘要
The main part of this thesis aims at establishing the functional equa- tion of Rankin L-functions over arbitrary global function fields k with odd characteristic. Fixing an " infinite" place ∞ of k, we consider an imaginary quadratic field 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 Rück-Tipp (functional equation over the rational func- tion fields ). We also derive a Wiener-Ikehara Tauberian theorem for global function fields in the last chapter for use in the analytic problems concerning arithmetic statistics. Asymptotic formulas for counting positive divisors of a given function field is investigated. This allows us to get in particular an asymptotic formula for the proportion of polynomials of a given degree over a finite field which do not have odd degree irreducible factors.