AI 中文总结
该研究在广义黎曼假设下，对固定权值素水平的全纯偶尖点新形式族的低位零点数，得出了新下界、正比例形式有零点及多项式上尾界等结果。
AI 中文摘要
我们研究固定权值与素水平的全纯偶尖点新形式族中的低位零点，特别关注在中心点附近指定归一化窗口内具有零点的形式数量，以及此类零点数量在形式间的分布。我们量化了在窗口内至少有一个零点的形式数量，并研究该窗口内零点数量在形式间的分布。在广义黎曼假设（Generalized Riemann Hypothesis）假设下，我们获得了具有低位零点的形式数量的新下界。我们首先证明，在无限的素水平序列N上，此类形式的数量为≫N^(7/8)log N。随后，我们利用高阶中心矩和合适的检验函数，证明该族中有正比例的形式在指定窗口内存在零点。最后，我们获得了该区域内零点数量的多项式上尾界，并证明有正比例的形式具有有界的非零低位零点数量。
英文摘要
We study low-lying zeros in families of even holomorphic cuspidal newforms of fixed weight and prime level, with particular emphasis on the number of forms having a zero in a prescribed normalized window about the central point and on the distribution of the number of such zeros among the forms. We quantify the number of forms having at least one zero in the window and study the distribution of the number of zeros in that window among the forms. Assuming the Generalized Riemann Hypothesis, we obtain new lower bounds for the number of forms having a low-lying zero. We first prove that, along an infinite sequence of prime levels $N$, the number of such forms is $\gg N^{7/8}\log N$. We then use higher centered moments and an appropriate test function to show that a positive proportion of the family has a zero in a prescribed window. Finally, we obtain polynomial upper-tail bounds for the number of zeros occurring there and show that a positive proportion of forms have a bounded, nonzero number of low-lying zeros.
