权为2的Siegel Eisenstein级数与四元代数  

Siegel Eisenstein series of weight 2 and quaternion algebra

作　　者：周海港[1,2] 李丁 Haigang Zhou;Ding Li

机构地区：[1]同济大学数学科学学院,上海200092 [2]河南省科学院数学研究所,郑州450046 [3]南昌航空大学数学与信息科学学院,南昌330063

出　　处：《中国科学：数学》2024年第9期1219-1244,共26页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:12271405)资助项目。

摘　　要：本文研究阶为2、权为2和级(level)为N为无平方因子的Siegel Eisenstein级数空间的维数与基,及它们与正定四元代数order的theta级数间的算术关系.首先,本文构造出级N为无平方因子时Siegel Eisenstein级数空间基的两种形式,并利用改进的Hurwitz类数给出Fourier系数的确切表达式;其次,对于Yoshida定义的关于有理数域Q上正定四元代数的Eichler order上的theta级数,本文得到了其Fourier系数明确表达式和Siegel-Weil型公式.In this paper,we consider the dimension and the basis of the space of the Siegel Eisenstein series of degree 2,weight 2,on the congruence subgroup of squarefree level N.We determine the arithmetic relation between the Siegel Eisenstein series and theta series associated with the order of the positive definite quaternion algebra.Firstly,we construct a full set of the basis of the space of squarefree level Siegel Eisenstein series and obtain their exact formulas for Fourier coefficients by using the modified Hurwitz class number.Secondly,for the theta series defined by Yoshida on the Eichler order of a positive definite quaternion algebra over the rational number field Q,we compute its Fourier coefficients and the Siegel-Weil type formula explicitly.

关 键 词：Siegel模形式 Eisenstein级数 正定四元代数 Eichler order 类数 theta级数 

分 类 号：O173[理学—数学]