论文标题
$ \ mathrm {gl}（2）$的强度多样性的改进
Refinements of strong multiplicity one for $\mathrm{GL}(2)$
论文作者
论文摘要
对于独特的单一cuspidal自动形式表示$π_1$和$π_2$，$ \ mathrm {gl}（2）$在数字$ f $上以及任何$α\ in \ bbb {r} $中的任何$α\，让$ \ mathcal { $λ_{π_1}（v）\ neq e^{iα}λ_{π_2}（v）$，其中$λ_{π_i}（v）$是$π_i$的傅立叶系数。在本文中，我们表明$ \ MATHCAL {s}_α$的下部dirichlet密度至少为$ \ frac {1} {16} $。此外，如果$π_1$和$π_2$不是扭曲等效的，我们表明，$ \ Mathcal {s}_α$和$ \cap_α\ Mathcal {s} s}_α$的较低dirichlet密度至少至少是$ \ frac {2} $ \ frac $ \ frac} $ \ frac {1}。此外，对于非扭转等效的$π_1$和$π_2$，如果每个$π_i$都对应于一个非CM重量$ k_i \ ge 2 $的非CM新形式，并且带有琐碎的Nebentypus，我们会获得各种上限，以获取Primes $ p \ le X $ p \ le x $ p \ le x $ p \ lec）$λ_____________________________________1 λ_{π_2}（p）^2 $。这些目前的改进了Murty-Pujahari，Murty-Rajan，Ramakrishnan和Walji的作品。
For distinct unitary cuspidal automorphic representations $π_1$ and $π_2$ for $\mathrm{GL}(2)$ over a number field $F$ and any $α\in\Bbb{R}$, let $\mathcal{S}_α$ be the set of primes $v$ of $F$ for which $λ_{π_1}(v)\neq e^{iα} λ_{π_2}(v)$, where $λ_{π_i}(v)$ is the Fourier coefficient of $π_i$ at $v$. In this article, we show that the lower Dirichlet density of $\mathcal{S}_α$ is at least $\frac{1}{16}$. Moreover, if $π_1$ and $π_2$ are not twist-equivalent, we show that the lower Dirichlet densities of $\mathcal{S}_α$ and $ \cap_α\mathcal{S}_α$ are at least $\frac{2}{13}$ and $\frac{1}{11}$, respectively. Furthermore, for non-twist-equivalent $π_1$ and $π_2$, if each $π_i$ corresponds to a non-CM newform of weight $k_i\ge 2$ and with trivial nebentypus, we obtain various upper bounds for the number of primes $p\le x$ such that $λ_{π_1}(p)^2 = λ_{π_2}(p)^2$. These present refinements of the works of Murty-Pujahari, Murty-Rajan, Ramakrishnan, and Walji.
