论文标题
立方PEL方程L功能
The cubic Pell equation L-function
论文作者
论文摘要
对于$ d> 1 $ cubefree有理整数，我们定义$ l $ function（表示为$ l_d（s）$），其系数源自$ \ mathbb q \ left（\ sqrt {-3}} \ right）的$ \ mathbb q \ left（\ sqrt Q \ left）的Cubic Theta函数。 The Dirichlet series defining $L_d(s)$ converges for $\text{Re}(s) > 1$, and its coefficients vanish except at values corresponding to integral solutions of $mx^3 - dny^3 = 1$ in $\mathbb Q\left(\sqrt{-3}\right)$, where $m$ and $n$ are squarefree.通过概括用于证明Takhtajan-Vinogradov Trace Formula的方法，我们获得了$ l_d（s）$至$ \ text {re}（s）> \ frac {1} {1} {2} {2} $的meromorphic延续| s |^{\ frac {7} {2}} $，并且在$ s = \ frac {2} {3} $上具有一个可能的简单极点，在某个Appell超几何功能的零处可能的极点，没有其他杆。我们猜想后一种情况没有发生，因此$ l_d（s）$没有其他电杆，带有$ \ text {re}（s）> \ frac {1} {2} {2} $，除了可能的简单极点$ s = \ frac {2} {3} {3} $。
For $d > 1$ a cubefree rational integer, we define an $L$-function (denoted $L_d(s)$) whose coefficients are derived from the cubic theta function for $\mathbb Q\left(\sqrt{-3}\right)$. The Dirichlet series defining $L_d(s)$ converges for $\text{Re}(s) > 1$, and its coefficients vanish except at values corresponding to integral solutions of $mx^3 - dny^3 = 1$ in $\mathbb Q\left(\sqrt{-3}\right)$, where $m$ and $n$ are squarefree. By generalizing the methods used to prove the Takhtajan-Vinogradov trace formula, we obtain the meromorphic continuation of $L_d(s)$ to $\text{Re}(s) > \frac{1}{2}$ and prove that away from its poles, it satisfies the bound $L_d(s) \ll |s|^{\frac{7}{2}}$ and has a possible simple pole at $s = \frac{2}{3}$, possible poles at the zeros of a certain Appell hypergeometric function, with no other poles. We conjecture that the latter case does not occur, so that $L_d(s)$ has no other poles with $\text{Re}(s) > \frac{1}{2}$ besides the possible simple pole at $s = \frac{2}{3}$.
