ON THE NON-TRIVIALITY OF THE $p$ -ADIC ABEL–JACOBI IMAGE OF GENERALISED HEEGNER CYCLES MODULO $p$ , II: SHIMURA CURVES
ON THE NON-TRIVIALITY OF THE $p$ -ADIC ABEL–JACOBI IMAGE OF GENERALISED HEEGNER CYCLES MODULO $p$ , II: SHIMURA CURVES
复制标题
论广义 Heegner 循环模 $p$ 的 $p$ -ADIC ABEL-JACOBI 图像的非平凡性，II：SHIMURA 曲线
DOI：
10.1017/s147474801500016x
复制
发表时间：
2015
影响因子：
0.9
通讯作者：
Ashay A. Burungale
中科院分区：
文献类型：
--
作者：
Ashay A. Burungale
广义Heegner圈与虚二次扩张$K/\mathbf{Q}$上的一对椭圆Newform和一个Hecke特征标相关联。周期生活在一个中维周组的久贺佐藤品种所产生的不确定志村曲线的理性和自我产品的CM阿贝尔表面。设$p$是在$K/\mathbf{Q}$中的奇素数分裂。本文证明了$K$的$\mathbf{Z}_{p}$ -反割圆扩张上模$p$广义Heegner圈的$p$ -adic Abel-Jacobi象的非平凡性。这个结果暗示了Chow群上coniveau滤子的最高阶段中广义Heegner圈的非平凡性，并证明了Mazur猜想的一个更高权的类比.在权重为2的情况下，该结果改进了Cornut-Vatsal和Aflalo-Nekovásness关于$K$的$\mathbf{Z}_{p}$ -反割圆扩张上Heegner点的非平凡性的结果.
Generalised Heegner cycles are associated to a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension $K/\mathbf{Q}$ . The cycles live in a middle-dimensional Chow group of a Kuga–Sato variety arising from an indefinite Shimura curve over the rationals and a self-product of a CM abelian surface. Let $p$ be an odd prime split in $K/\mathbf{Q}$ . We prove the non-triviality of the $p$ -adic Abel–Jacobi image of generalised Heegner cycles modulo $p$ over the $\mathbf{Z}_{p}$ -anticyclotomic extension of $K$ . The result implies the non-triviality of the generalised Heegner cycles in the top graded piece of the coniveau filtration on the Chow group, and proves a higher weight analogue of Mazur’s conjecture. In the case of weight 2, the result provides a refinement of the results of Cornut–Vatsal and Aflalo–Nekovář on the non-triviality of Heegner points over the $\mathbf{Z}_{p}$ -anticyclotomic extension of $K$ .