发表机构
Soochow University; Westlake University（苏州大学; 西湖大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在GRH下研究二次Hecke L函数在导子方面的极值，证明特定点处存在大值，并给出达到较小常数的特征数量下界，同时推广到函数域并给出固定条带及s=1处的界。
AI 中文摘要
我们研究二次Hecke $L$-函数在导子方面的极大值。设$K$为一个固定的数域，并假设其有限阶Hecke $L$-函数的GRH成立。在导子范数与$X$相当的固定射线类分量中，我们证明对每个固定的$A\geq0$，有\\[ \max_\chi L\left(\frac12+\frac A{\log_2X},\chi\right) \geq\exp\left\{(e^{-A}+o(1)) \sqrt{\frac{\log X\log_3X}{\log_2X}}\right\} \\]。每个固定的更小常数至少被$X^{1-o(1)}$个特征达到。同样的计数在适当缓慢移动并趋近于所示常数的阈值处也成立。组合输入是一个基数为$N$的稀疏无平方Gál集，保持已知的首项常数$2$，且具有平方乘法能量$N^{2+o(1)}$。该能量界反映了相关布尔多项式的低次数。结合共振估计，它给出了丰度界。我们还对奇特征全局函数域上避开一个固定位点的素数导子的二次特征证明了无条件类似结果。最后，我们给出了固定条带及$s=1$处的界，包括对Dedekind zeta函数留数和指定局部因子的依赖。
英文摘要
We study large values of quadratic Hecke $L$-functions in the conductor aspect. Let $K$ be a fixed number field, and assume GRH for its finite-order Hecke $L$-functions. In a fixed ray class component with conductor norm comparable to $X$, we prove that \[ \max_χL\left(\frac12+\frac A{\log_2X},χ\right) \geq\exp\left\{(e^{-A}+o(1)) \sqrt{\frac{\log X\log_3X}{\log_2X}}\right\} \] for every fixed $A\geq0$. Every fixed smaller constant is attained by at least $X^{1-o(1)}$ characters. The same count holds at a suitably slowly moving threshold approaching the displayed constant. The combinatorial input is a sparse squarefree Gál set of cardinality $N$, retaining the known leading constant $2$ and having square multiplicative energy $N^{2+o(1)}$. The energy bound reflects the low degree of the associated Boolean polynomial. Together with the resonance estimate, it yields the abundance bound. We also prove unconditional analogues for quadratic characters with prime conductor away from one fixed place over any global function field of odd characteristic. Finally, we give bounds in the fixed strip and at $s=1$, including the dependence on the residue of the Dedekind zeta function and the prescribed local factors.
Comments42 pages
