AI 中文总结
该研究针对椭圆曲线 $y^2=x^3-dx$ 的 $L$ 函数族，在广义黎曼假设及相关猜想下，分别得到平均解析秩上界 $\frac{13}{6}$ 和 $\frac{3}{2}$，并证明存在正比例扭曲线的解析秩为0或1。
AI 中文摘要
我们研究当 $d$ 取遍无四次因子的奇数时，与椭圆曲线 $E_d: y^2=x^3-dx$ 相关的 $L$ 函数族 $L(s, E_d)$ 的平均解析秩。由于这是具有复乘法的曲线族，故有 $L(s, E_d)=L(s - \frac{1}{2}, \xi_d)$，其中 $\xi_d$ 是 $\mathbb{Z}[i]$ 上的赫克特征。在假设广义黎曼假设成立的前提下，我们对傅里叶变换支撑在 $(-\frac{3}{5}, \frac{3}{5})$ 内的测试函数，计算该族低阶零点的一级密度。由此得到该族平均解析秩 $r(E_d)$ 的上界为 $\frac{13}{6}$。在额外假设素元处四次高斯和分布的猜想（即三次高斯和的帕特森猜想的四次类似物）成立的条件下，我们将容许支撑扩展至 $(-1, 1)$，并将平均解析秩的上界改进为 $\frac{3}{2}$。两个结果均表明，存在正比例的扭曲线满足 $r(E_d)=1$，而第二个结果还给出了满足 $r(E_d)=0$ 的正比例扭曲线。
英文摘要
We study the average analytic rank in the family of $L$-functions $L(s, E_d)$ associated with the elliptic curves $E_d : y^2=x^3-dx$, as $d$ varies over fourth-power-free odd integers. Since this is a family of curves with complex multiplication, we have $L(s, E_d)=L(s - \frac12, ξ_d)$, where $ξ_d$ is a Hecke character over $\mathbb{Z}[i]$. Assuming the Generalized Riemann Hypothesis, we compute the one-level density of the low-lying zeros of this family for test functions whose Fourier transform is supported in $(-\frac35, \frac35)$. As a consequence, we obtain the upper bound $\frac{13}{6}$ for the average analytic rank $r(E_d)$ over the family. Under the additional assumption of a conjecture on the distribution of quartic Gauss sums at prime elements (a quartic analogue of Patterson's conjecture for cubic Gauss sums), we extend the admissible support to $(-1, 1)$ and improve the upper bound for the average analytic rank to $\frac32$. Both results imply that a positive proportion of twists satisfy $r(E_d) =1$, while the second also yields a positive proportion of twists with $r(E_d)=0$.
