论文标题
模块化符号和同一个符号类的剩余等分分配给双曲线$ n $ -space的商类
Residual equidistribution of modular symbols and cohomology classes for quotients of hyperbolic $n$-space
论文作者
论文摘要
我们使用Eisenstein系列提供了一种新的简单的自动形态方法，以研究模块化符号Modulo Primes的等分分配，我们将其应用于Mazur和Rubin的平均版本。更确切地说，我们证明，与重量2尖的hecke基础相对应的模块化符号是渐近地共同等级的mod $ p $，而我们允许对牙孔的位置进行限制。作为应用程序，我们获得了Dedekind总和的残差等分结果。此外，我们计算分布的方差，并与扰动理论的联系表现出令人惊讶的偏见。此外，在某些特定情况下，我们使用与爱森斯坦一致性的联系证明了全部猜想。最后，我们的方法概括为$ n $维超薄空间的有限体积商的共同分配结果。
We provide a new and simple automorphic method using Eisenstein series to study the equidistribution of modular symbols modulo primes, which we apply to prove an average version of a conjecture of Mazur and Rubin. More precisely, we prove that modular symbols corresponding to a Hecke basis of weight 2 cusp forms are asymptotically jointly equidistributed mod $p$ while we allow restrictions on the location of the cusps. As an application, we obtain a residual equidistribution result for Dedekind sums. Furthermore, we calculate the variance of the distribution and show a surprising bias with connections to perturbation theory. Additionally, we prove the full conjecture in some particular cases using a connection to Eisenstein congruences. Finally, our methods generalise to equidistribution results for cohomology classes of finite volume quotients of $n$-dimensional hyperbolic space.
