阿贝尔曲面的约化例外分裂
数论
2020-03-18 v3 代数几何
摘要
基于 Sato–Tate 猜想的启发式推理表明，定义在数域上的阿贝尔曲面有无穷多个分裂约化的位。我们证明了具有实乘法的阿贝尔曲面的这一结果。类似于 Charles 关于给定椭圆曲线对的约化例外同源的定理以及 Elkies 关于给定椭圆曲线的超奇异约化的定理，我们的定理表明，与阿贝尔簇约化相关的密度为零的素数集合是无穷的。证明依赖于 Hilbert 模曲面上的 Arakelov 相交理论。
引用
@article{arxiv.1706.08154,
title = {Exceptional splitting of reductions of abelian surfaces},
author = {Ananth N. Shankar and Yunqing Tang},
journal= {arXiv preprint arXiv:1706.08154},
year = {2020}
}
备注
18 pages. We generalize the main result in v1 (only for abelian surfaces over the field of rationals) to the case of abelian surfaces with real multiplication over any number fields. This improvement is due to the new treatment of finite contribution in arithmetic intersection number in section 3