论文标题
椭圆曲线曲折的家族的一级密度在功能场上
One-level density of the family of twists of an elliptic curve over function fields
论文作者
论文摘要
我们修复了椭圆曲线$ e/\ mathbb {f} _q（t）$，并考虑$ \ {e \otimesχ_d\} $ of $ e $ of quadratic dirichlet字符扭曲的$ e $。证明其$ L $ functions的一级密度遵循正交对称性的测试功能，并在$（-1,1）$中支持傅立叶变换。作为应用程序，我们在平均分析等级上获得了3/2的上限。通过按照功能方程式的迹象分开家庭，我们获得了至少$ 12.5 \％的家庭排名为零，至少$ 37.5 \％$ $排名第一。 Katz和Sarnak哲学预测，这些百分比都应为$ 50 \％$，平均分析等级应为$ 1/2 $。我们通过计算订单$ \ ell \ ell \ neq 2 $ coprime的dirichlet字符的$ e $ twist的一级密度来完成。我们通过单一对称性获得$（1/2,1/2）$的限制。
We fix an elliptic curve $E/\mathbb{F}_q(t)$ and consider the family $\{E\otimesχ_D\}$ of $E$ twisted by quadratic Dirichlet characters. The one-level density of their $L$-functions is shown to follow orthogonal symmetry for test functions with Fourier transform supported inside $(-1,1)$. As an application, we obtain an upper bound of 3/2 on the average analytic rank. By splitting the family according to the sign of the functional equation, we obtain that at least $12.5\%$ of the family have rank zero, and at least $37.5\%$ have rank one. The Katz and Sarnak philosophy predicts that those percentages should both be $50\%$ and that the average analytic rank should be $1/2$. We finish by computing the one-level density of $E$ twisted by Dirichlet characters of order $\ell\neq 2$ coprime to $q$. We obtain a restriction of $(-1/2,1/2)$ on the support with a unitary symmetry.
