论文标题
关于Hoffstein和Kontorovich的问题
On a problem of Hoffstein and Kontorovich
论文作者
论文摘要
让$π$为$ \ operatotorname {gl} _2（\ mathbb {a} _ {\ mathbb {q}}}）$和$ d $的cuspidal自动形态表示。 Hoffstein和Kontorovich要求在最少$ | d | $（如果存在的话）上进行绑定，以使中心值$ l（1/2，π\ otimesχ_d）\ neq 0 $。绑定应根据重量，拉普拉斯特征值和/或$π$的水平给出。 让$ f $为均匀重量$ \ ell $，奇数cubefree级别$ n $和琐碎的Nebentypus的全态扭曲最小新形式。当$π\congπ_f$和$ n $的无方面部分的大小适当时，我们有条件地改善了霍夫斯坦和kontorovich的水平级别的结果（在subconvexity（带有子范围指数）的自动形态$ l $ rungunctions）下。结果，我们有条件地证明，给定椭圆曲线$ e/\ mathbb {q} $的导体$ n $，存在一个带有mordell的小扭曲 - 韦尔等级等于零。
Let $π$ be a cuspidal automorphic representation of $\operatorname{GL}_2(\mathbb{A}_{\mathbb{Q}})$ and $d$ be a fundamental discriminant. Hoffstein and Kontorovich ask for a bound on the least $|d|$ (if it exists) such that the central value $L(1/2, π\otimes χ_d) \neq 0$. The bound should be given in terms of the weight, Laplace eigenvalue and/or level of $π$. Let $f$ be a holomorphic twist-minimal newform of even weight $\ell$, odd cubefree level $N$, and trivial nebentypus. When $π\cong π_f$ and the squarefree part of $N$ is of appropriate size, we conditionally improve upon level aspect results of Hoffstein and Kontorovich under subconvexity (with a sub-Weyl exponent) for automorphic $L$-functions. As a consequence we conditionally prove that given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, there exists a small twist that has Mordell--Weil rank equal to zero.
