有限域上循环三次曲线的迹的统计
数论
2010-06-17 v2
摘要
我们研究了当曲线在模空间的不可约分量中变化时,与q元域上亏格为g的循环三次曲线相关联的Frobenius自同态迹的变化。我们证明,对于固定的q和递增的g,Frobenius迹的极限分布等于q+1个独立随机变量之和,这些随机变量取值0的概率为2/(q+2),取值1、e^{(2pi i)/3}、e^{(4pi i)/3}的概率各为q/(3(q+2))。这推广了Kurlberg和Rudnick关于超椭圆曲线相同极限的工作。我们还证明,当g和q都趋于无穷时,归一化迹具有标准复高斯分布,并展示了如何将这些结果推广到射影直线的p重覆盖。
引用
@article{arxiv.0907.5434,
title = {Statistics for traces of cyclic trigonal curves over finite fields},
author = {Alina Bucur and Chantal David and Brooke Feigon and Matilde Lalín},
journal= {arXiv preprint arXiv:0907.5434},
year = {2010}
}
备注
30 pages, added statement and sketch of proof in Section 7 for generalization of results to p-fold covers of the projective line, the final version of this article will be published in International Mathematics Research Notices