论文标题
超越扩展的Selberg类：$ d_f \ le 1 $
Beyond the extended Selberg class: $d_F\le 1$
论文作者
论文摘要
我们将介绍两个新类的Dirichlet系列，它们在乘法下是单型。第一类$ \ Mathfrak {a}^{\＃} $都包含Kaczorowski和Perelli的扩展名Selberg类$ \ Mathscr {s}^{\＃} $，以及许多$ l $ lub-functions automorphic代表$ { $ {\ mathbb a} _k $表示数字字段$ k $上的adèles（这些表示不必统一或通用）。这与类$ \ mathscr {s}^{\＃} $相反，该类别较小并且已知包含，很少有这些$ l $ functions。较大的类是通过削弱绝对收敛的需求，允许有限数量的杆子，允许更通用的伽马因子，并允许该系列在$ \ mathrm {re}（s）= 1/2 $的权利的权利中，同时保留了Extended Selberg类的其他Axioms。我们将在$ \ mathfrak {a}^{\＃} $ $ d $时分类为$ d \ le 1 $（当$ d = 1 $时，我们将假设在$ \ mathrm {re}（s）> 1 $中的绝对收敛。我们将进一步证明$ {\ rm gl} _2（{\ Mathbb a} _ {\ mathbb q}）$的$ l $ l $ functions $ {\ rm gl} _2（\ rm gl} _2（{\ mathbb a} _ {\ mathbb a}）$和一个定理，允许我们比较$ l $ l $ l $ - functions $ - functions的{ a} _K）$，无法从先前的分类结果中推导。 The second class $\mathfrak{G}^{\#}\subset\mathfrak{A}^{\#}$, which also contains $\mathscr{S}^{\#}$, more closely models the behaviour of $L$-functions of unitary globally generic representations of ${\rm GL}_n({\mathbb a} _k）$。
We will introduce two new classes of Dirichlet series which are monoids under multiplication. The first class $\mathfrak{A}^{\#}$ contains both the extended Selberg class $\mathscr{S}^{\#}$ of Kaczorowski and Perelli as well as many $L$-functions attached to automorphic representations of ${\rm GL}_n({\mathbb A}_K)$, where ${\mathbb A}_K$ denotes the adèles over the number field $K$ (these representations need not be unitary or generic). This is in contrast to the class $\mathscr{S}^{\#}$ which is smaller and is known to contain, very few of these $L$-functions. The larger class is obtained by weakening the requirement for absolute convergence, allowing a finite number of poles, allowing more general gamma factors and by allowing the series to have trivial zeros to the right of $\mathrm{Re}(s)=1/2$, while retaining the other axioms of the extended Selberg class. We will classify series in $\mathfrak{A}^{\#}$ of degree $d$ when $d\le 1$ (when $d=1$, we will assume absolute convergence in $\mathrm{Re}(s)>1$). We will further prove a primitivity result for the $L$-functions of cuspidal eigenforms on ${\rm GL}_2({\mathbb A}_{\mathbb Q})$ and a theorem allowing us to compare the zeros of tensor product $L$-functions of ${\rm GL}_n({\mathbb A}_K)$ which cannot be deduced from previous classification results. The second class $\mathfrak{G}^{\#}\subset\mathfrak{A}^{\#}$, which also contains $\mathscr{S}^{\#}$, more closely models the behaviour of $L$-functions of unitary globally generic representations of ${\rm GL}_n({\mathbb A}_K)$.
