We define the notion of special automorphisms on Shimura curves. Using this notion, for a wild class of elliptic curves defined over Q, we get rank one quadratic twists by discriminants having any prescribed number of...We define the notion of special automorphisms on Shimura curves. Using this notion, for a wild class of elliptic curves defined over Q, we get rank one quadratic twists by discriminants having any prescribed number of prime factors. Finally, as an application, we obtain some new results on Birch and Swinnerton-Dyer(BSD) conjecture for the rank one quadratic twists of the elliptic curve X_0(49).展开更多
This is the note for a series of lectures that the author gave at the Centre de Recerca Matemtica (CRM), Bellaterra, Barcelona, Spain on October 19–24, 2009. The aim is to give a comprehensive description of some rec...This is the note for a series of lectures that the author gave at the Centre de Recerca Matemtica (CRM), Bellaterra, Barcelona, Spain on October 19–24, 2009. The aim is to give a comprehensive description of some recent work of the author and his students on generalisations of the Gross-Zagier formula, Euler systems on Shimura curves, and rational points on elliptic curves.展开更多
We prove that a Shimura curve in the Siegel modular variety is not generically contained in the open Torelli locus as long as the rank of unitary part in its canonical Higgs bundle satisfies a numerical upper bound.As...We prove that a Shimura curve in the Siegel modular variety is not generically contained in the open Torelli locus as long as the rank of unitary part in its canonical Higgs bundle satisfies a numerical upper bound.As an application we show that the Coleman–Oort conjecture holds for Shimura curves associated with partial corestriction upon a suitable choice of parameters,which generalizes a construction due to Mumford.展开更多
The motivation for this paper is the study of arithmetic properties of Shimura varieties,in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level struct...The motivation for this paper is the study of arithmetic properties of Shimura varieties,in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure.This is closely related to the structure of Rapoport–Zink spaces and of affine Deligne–Lusztig varieties.We prove a Hodge–Newton decomposition for affine Deligne–Lusztig varieties and for the special fibers of Rapoport–Zink spaces,relating these spaces to analogous ones defined in terms of Levi subgroups,under a certain condition(Hodge–Newton decomposability)which can be phrased in combinatorial terms.Second,we study the Shimura varieties in which every non-basic risogeny classis Hodge–Newton decomposable.We show that(assuming the axioms of He and Rapoport in Manuscr.Math.152(3–4):317–343,2017)this condition is equivalent to nice conditions on either the basic locus or on all the non-basic Newton strata of the Shimura varieties.We also give a complete classification of Shimura varieties satisfying these conditions.While previous results along these lines often have restrictions to hyperspecial(or at least maximal parahoric)level structure,and/or quasi-split underlying group,we handle the cases of arbitrary parahoric level structure and of possibly non-quasisplit underlying groups.This results in a large number of new cases of Shimura varieties where a simple description ofthe basiclocus can be expected.As a striking consequence of the results,we obtain that this property is independent of the parahoric subgroup chosen as level structure.We expect that our conditions are closely related to the question whether the weakly admissible and admissible loci coincide.展开更多
The parameterization of the elliptic curve Y2=X2+1 is given by using the modular forms of X(12) Then using class field theory over imaginary quadratic field and Shimura reciprocity, an infinite order point on the curv...The parameterization of the elliptic curve Y2=X2+1 is given by using the modular forms of X(12) Then using class field theory over imaginary quadratic field and Shimura reciprocity, an infinite order point on the curve y2- = x3-p3 is constructed, for prime p=7 (mod 24).展开更多
We shall give a simple （basically） the Igusa tower over Shimura varieties of PEL purely in characteristic p proof of the irreducibility of type. Our result covers Shimura variety of type A and type C classical group...We shall give a simple （basically） the Igusa tower over Shimura varieties of PEL purely in characteristic p proof of the irreducibility of type. Our result covers Shimura variety of type A and type C classical groups, in particular, the Siegel modular varieties, the Hilbert-Siegel modular varieties, Picard surfaces and Shimura varieties of inner forms of unitary and symplectic groups over totally real fields.展开更多
文摘We define the notion of special automorphisms on Shimura curves. Using this notion, for a wild class of elliptic curves defined over Q, we get rank one quadratic twists by discriminants having any prescribed number of prime factors. Finally, as an application, we obtain some new results on Birch and Swinnerton-Dyer(BSD) conjecture for the rank one quadratic twists of the elliptic curve X_0(49).
文摘This is the note for a series of lectures that the author gave at the Centre de Recerca Matemtica (CRM), Bellaterra, Barcelona, Spain on October 19–24, 2009. The aim is to give a comprehensive description of some recent work of the author and his students on generalisations of the Gross-Zagier formula, Euler systems on Shimura curves, and rational points on elliptic curves.
基金supported by SFB/Transregio 45 Periods,Moduli Spaces and Arithmetic of Algebraic Varieties of DFG,by NSF of China Grant Nos.11771203,11231003,11301495Fundamental Research Funds for the Central Universities,Nanjing University,No.0203-14380009by the Science Foundation of Shanghai(No.13DZ2260400).
文摘We prove that a Shimura curve in the Siegel modular variety is not generically contained in the open Torelli locus as long as the rank of unitary part in its canonical Higgs bundle satisfies a numerical upper bound.As an application we show that the Coleman–Oort conjecture holds for Shimura curves associated with partial corestriction upon a suitable choice of parameters,which generalizes a construction due to Mumford.
基金supported by DFG Transregio-Sonderforschungsbereich 45Xuhua He was partially supported by NSF DMS-1463852Sian Nie was partially supported by NSFC(Nos.11501547 and 11688101)and QYZDB-SSW-SYS007.
文摘The motivation for this paper is the study of arithmetic properties of Shimura varieties,in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure.This is closely related to the structure of Rapoport–Zink spaces and of affine Deligne–Lusztig varieties.We prove a Hodge–Newton decomposition for affine Deligne–Lusztig varieties and for the special fibers of Rapoport–Zink spaces,relating these spaces to analogous ones defined in terms of Levi subgroups,under a certain condition(Hodge–Newton decomposability)which can be phrased in combinatorial terms.Second,we study the Shimura varieties in which every non-basic risogeny classis Hodge–Newton decomposable.We show that(assuming the axioms of He and Rapoport in Manuscr.Math.152(3–4):317–343,2017)this condition is equivalent to nice conditions on either the basic locus or on all the non-basic Newton strata of the Shimura varieties.We also give a complete classification of Shimura varieties satisfying these conditions.While previous results along these lines often have restrictions to hyperspecial(or at least maximal parahoric)level structure,and/or quasi-split underlying group,we handle the cases of arbitrary parahoric level structure and of possibly non-quasisplit underlying groups.This results in a large number of new cases of Shimura varieties where a simple description ofthe basiclocus can be expected.As a striking consequence of the results,we obtain that this property is independent of the parahoric subgroup chosen as level structure.We expect that our conditions are closely related to the question whether the weakly admissible and admissible loci coincide.
基金the National Natural Science Foundation of China
文摘The parameterization of the elliptic curve Y2=X2+1 is given by using the modular forms of X(12) Then using class field theory over imaginary quadratic field and Shimura reciprocity, an infinite order point on the curve y2- = x3-p3 is constructed, for prime p=7 (mod 24).
基金supported by the NSF grants:DMS 0244401,DMS 0456252DMS 0753991
文摘We shall give a simple （basically） the Igusa tower over Shimura varieties of PEL purely in characteristic p proof of the irreducibility of type. Our result covers Shimura variety of type A and type C classical groups, in particular, the Siegel modular varieties, the Hilbert-Siegel modular varieties, Picard surfaces and Shimura varieties of inner forms of unitary and symplectic groups over totally real fields.
