Let f denote a normalized Maass cusp form for SL（2, Z）, which is an eigenfunction of all the Hecke operators T（n） as well as the reflection operator T-1： z →z. We obtain a zero-density result of the L-function a...Let f denote a normalized Maass cusp form for SL（2, Z）, which is an eigenfunction of all the Hecke operators T（n） as well as the reflection operator T-1： z →z. We obtain a zero-density result of the L-function attached to f near σ = 1. This improves substantially the previous results in this direction.展开更多
In this paper,we present a very simple explicit description of Langlands Eisenstein series for SL(n,Z).The functional equations of these Eisenstein series are heuristically derived from the functional equations of cer...In this paper,we present a very simple explicit description of Langlands Eisenstein series for SL(n,Z).The functional equations of these Eisenstein series are heuristically derived from the functional equations of certain divisor sums and certain Whittaker functions that appear in the Fourier coefficients of the Eisenstein series.We conjecture that the functional equations are unique up to a real affine transformation of the s variables defining the Eisenstein series and prove the uniqueness conjecture in certain cases.展开更多
It is well known by the strong multiplicity one thatπis uniquely determined by the Satake parameter c(π,v)for almost all v.Also,it suffices for us to test only finitely many v.We proved some S-effective version of m...It is well known by the strong multiplicity one thatπis uniquely determined by the Satake parameter c(π,v)for almost all v.Also,it suffices for us to test only finitely many v.We proved some S-effective version of multiplicity one theorems.Roughly speaking,ifπandπ′are not equivalent,then there is also a bound N(S)which is some expression in terms of K,d and max(N(π),N(π′)),which are analytic conductor ofπandπ′,respectively(will be defined soon),such that there is a v/∈S withπv～=π′vand N pv<N.We also proved S-effective multiplicity one for the Chebotarev Density Theorem,and for GL(1).展开更多
文摘Let f denote a normalized Maass cusp form for SL（2, Z）, which is an eigenfunction of all the Hecke operators T（n） as well as the reflection operator T-1： z →z. We obtain a zero-density result of the L-function attached to f near σ = 1. This improves substantially the previous results in this direction.
基金supported by Simons Collaboration(Grant No.567168)。
文摘In this paper,we present a very simple explicit description of Langlands Eisenstein series for SL(n,Z).The functional equations of these Eisenstein series are heuristically derived from the functional equations of certain divisor sums and certain Whittaker functions that appear in the Fourier coefficients of the Eisenstein series.We conjecture that the functional equations are unique up to a real affine transformation of the s variables defining the Eisenstein series and prove the uniqueness conjecture in certain cases.
基金supported by the State Key Development Program for Basic Researchof China(973 project)(Grant No.2013CB834202)National Natural Science Foundation of China(Grant No.11321101)the One Hundred Talent’s Program from Chinese Academy of Science
文摘It is well known by the strong multiplicity one thatπis uniquely determined by the Satake parameter c(π,v)for almost all v.Also,it suffices for us to test only finitely many v.We proved some S-effective version of multiplicity one theorems.Roughly speaking,ifπandπ′are not equivalent,then there is also a bound N(S)which is some expression in terms of K,d and max(N(π),N(π′)),which are analytic conductor ofπandπ′,respectively(will be defined soon),such that there is a v/∈S withπv～=π′vand N pv<N.We also proved S-effective multiplicity one for the Chebotarev Density Theorem,and for GL(1).
