论文标题
Zeta函数和非趋化定理在$ \ mathrm {gl} _2 $上
Zeta functions and nonvanishing theorems for toric periods on $\mathrm{GL}_2$
论文作者
论文摘要
让$ f $为一个数字字段，$ d $ a xaternion代数超过$ f $。使用cuspidal自动形式表示$ d_ \ mathbb {a}^\ times $带有琐碎的central cahracter。我们研究Zeta功能，具有$π$的周期积分，用于perhomensoodos vector Space $（d^\ times \ times d^\ times \ times \ times \ times \ mathrm {gl} _2，d \ oplus d）$。我们显示了它们的Meromormormorphic延续和功能方程，确定可能的杆的位置和顺序并计算残基。沿着斋藤理论和计算未受到的局部因素的争论，获得了Zeta函数的明确公式。计算此明确公式的可能性杆的顺序，我们表明，如果$ l（1/2，π）\ neq0 $，则有无限的许多二次扩展$ e $ $ f $的$ f $嵌入$ d $，因此$π$与$ e $相对于$ e $ a $。
Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a cuspidal automorphic representation $π$ of $D_\mathbb{A}^\times$ with trivial central cahracter. We study the zeta functions with period integrals on $π$ for the perhomogeneous vector space $(D^\times\times D^\times\times\mathrm{GL}_2, D\oplus D)$. We show their meromorphic continuation and functional equation, determine the location and orders of possible poles and compute the residue. Arguing along the theory of Saito and computing unramified local factors, the explicit formula of the zeta functions is obtained. Counting the order of possible poles of this explicit formula, we show that if $L(1/2, π)\neq0$, there are infinitely many quadratic extension $E$ of $F$ which embeds in $D$, such that $π$ has nonvanishing toric period with respect to $E$.
