虚二次域上GL(2)的自守符号与自守L-值
Automorphic symbols and automorphic \emph{L}-values of GL(2) over imaginary quadratic fields
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中文总结 AI 辅助
该研究将Kim–Sun的同调策略从经典模曲线推广到算术轨形，证明虚二次域上平行权2 Hecke本征形式经Coates–Wiles特征扭转的L-函数临界值，在正密度素数集合上模素非零的特征占比为正。
中文摘要 AI 辅助
我们研究由Coates–Wiles特征扭转的虚二次域上L-函数临界值的模素非零性。对于虚二次域上的平行权2 Hecke本征形式，我们证明：存在一个正密度的素数集合，使得对其中每个素数，都有正比例的Coates–Wiles特征对应的整数值L-值模该素数非零。该论证构造了带整系数的抛物同调中的自守符号，并通过它们与抛物上同调类的配对来表示临界值。这些配对的垂直加性平均族可恢复傅里叶系数，并使符号生成的模具有满秩。我们将该满秩性质转化为模素非零性。这将Kim–Sun的同调策略从经典模曲线推广到算术轨形，同时解决了虚二次域情形特有的单位障碍。
英文摘要
We study non-vanishing modulo primes of critical values of $L$-functions over imaginary quadratic fields twisted by Coates--Wiles characters. For a parallel weight two Hecke eigenform over an imaginary quadratic field, we prove that a positive proportion of Coates--Wiles characters have nonzero integral $L$-values modulo each prime in a positive-density set. The argument constructs automorphic symbols in parabolic homology with integral coefficients and expresses the critical values through their pairings with parabolic cohomology classes. A vertical family of additive averages of these pairings recovers Fourier coefficients and force the module generated by symbols to have full rank. We transfer this full-rank property to non-vanishing modulo primes. This extends the homological strategy of Kim--Sun from classical modular curves to arithmetic orbifolds, while addressing the unit obstructions specific to the imaginary quadratic setting.
发表机构
- Ulsan National Institute of Science and Technology（蔚山科学技术院）
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