This paper reviews a less known rational structure on the Siegel modular variety X（N） = F（N）／Hg over Q for integers g, N ≥ 1. The author then describes explicitly how Galois groups act on CM points on this varie...This paper reviews a less known rational structure on the Siegel modular variety X（N） = F（N）／Hg over Q for integers g, N ≥ 1. The author then describes explicitly how Galois groups act on CM points on this variety. Finally, another proof of the Shimura reciprocity law by using the result and the q-expansion principle is given.展开更多
The notion of Higgs-de Rham flows was introduced by Lan et al.(2019),as an analogue of Yang-Mills-Higgs flows in the complex nonabelian Hodge theory.In this paper we investigate a small part of this theory,and study t...The notion of Higgs-de Rham flows was introduced by Lan et al.(2019),as an analogue of Yang-Mills-Higgs flows in the complex nonabelian Hodge theory.In this paper we investigate a small part of this theory,and study those Higgs-de Rham flows which are of level zero.We improve the original definition of level-zero Higgs-de Rham flows(which works for general levels),and establish a Hitchin-Simpson type correspondence between such objects and certain representations of fundamental groups in positive characteristic,which generalizes a classical results of Katz(1973).We compare the deformation theories of two sides in the correspondence,and translate the Galois action on the geometric fundamental groups of algebraic varieties defined over finite fields into the Higgs side.展开更多
基金supported by the National Science Foundation Grants(No.DMS-1200380,DMS-1500743)the Chinese Qian Ren Plan of Tsinghua(No.543100001)
文摘This paper reviews a less known rational structure on the Siegel modular variety X（N） = F（N）／Hg over Q for integers g, N ≥ 1. The author then describes explicitly how Galois groups act on CM points on this variety. Finally, another proof of the Shimura reciprocity law by using the result and the q-expansion principle is given.
基金supported by National Natural Science Foundation of China(Grant Nos.11622109 and 11721101)Anhui Initiative in Quantum Information Technologies(Grant No.AHY150200)supported by One-Thousand-Talents Program of China。
文摘The notion of Higgs-de Rham flows was introduced by Lan et al.(2019),as an analogue of Yang-Mills-Higgs flows in the complex nonabelian Hodge theory.In this paper we investigate a small part of this theory,and study those Higgs-de Rham flows which are of level zero.We improve the original definition of level-zero Higgs-de Rham flows(which works for general levels),and establish a Hitchin-Simpson type correspondence between such objects and certain representations of fundamental groups in positive characteristic,which generalizes a classical results of Katz(1973).We compare the deformation theories of two sides in the correspondence,and translate the Galois action on the geometric fundamental groups of algebraic varieties defined over finite fields into the Higgs side.
