论文标题
$ \ mathbb {h}^3 $中模块化符号的分布
Distribution of modular symbols in $\mathbb{H}^3$
论文作者
论文摘要
我们引入了一种用于研究模块化符号分布的新技术，我们将其应用于Bianchi组的一致性亚组。 We prove that if $K$ is a quadratic imaginary number field of class number one and $\mathcal{O}_K$ its ring of integers, then for certain congruence subgroups of $\mathrm{PSL}_2(\mathcal{O}_K)$, the periods of a cusp form of weight two obey asymptotically a normal distribution.这些结果是从我们方法适用的cofinite kleinian群体的商表面的更一般环境中进行的专业。我们避免时刻的方法。我们的新见解是使用Laplacian最小的特征值的行为，用于由模块化符号扭曲的空间。我们的方法还恢复了分布的第一刻和第二刻。
We introduce a new technique for the study of the distribution of modular symbols, which we apply to congruence subgroups of Bianchi groups. We prove that if $K$ is a quadratic imaginary number field of class number one and $\mathcal{O}_K$ its ring of integers, then for certain congruence subgroups of $\mathrm{PSL}_2(\mathcal{O}_K)$, the periods of a cusp form of weight two obey asymptotically a normal distribution. These results are specialisations from the more general setting of quotient surfaces of cofinite Kleinian groups, where our methods apply. We avoid the method of moments. Our new insight is to use the behaviour of the smallest eigenvalue of the Laplacian for spaces twisted by modular symbols. Our approach also recovers the first and the second moment of the distribution.
