Modular forms associated to real quadratic fields
Modular forms associated to real quadratic fields
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与实二次域相关的模形式
DOI：
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发表时间：
1975
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通讯作者：
D. Zagier
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文献类型：
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作者：
D. Zagier
本文的目的是构造SL 27 Z（及其某些同余子群）和真实的二次域的Hilbert模群的模形式。在w 1中，我们固定一个真实的二次域K，且k > 2，并构造了一系列函数ω 1，（Z1，Z2）（m=0，1，2，. . .)它们是Hilbert模群SL 2（9（9= K中的整数环）的权为k的模形式。形式co o是K的Hecke-Eisenstein级数的倍数，而所有其他的co，。是尖点形式。的傅立叶展开。（z 1，z2）在w 2中计算;每个傅立叶系数表示为无限和，其典型项是有限指数和（类似于Kloosterman和）与阶k 1的贝塞尔函数的乘积。主要结果是，对于上半平面.~中的任意点z1和z2，数m k-1 ω i，（z 1，z 2）（m= 1，2，.）是权重k的模形式（在一个变量中）的傅立叶系数。更准确地说，设D是K的判别式，e.=（D/）是K的特征标，S（D，k，~）是特征标为e的Fo（D）的权为k的尖点型空间，则对固定的z1，功能
The purpose of this paper is to construct modular forms, both for SL27Z (and certain of its congruence subgroups) and for the Hilbert modular group of a real quadratic field. In w 1 we fix a real quadratic field K and even integer k > 2 and construct a series of functions co,,(Zl, z2) (m=0, 1, 2, . . .) which are modular forms of weight k for the Hilbert modular group SL2(9 ((9=ring of integers in K). The form co o is a multiple of the Hecke-Eisenstein series for K, while all of the other co,. are cusp forms. The Fourier expansion of co,. (z 1, z2) is calculated in w 2; each Fourier coefficient is expressed as an infinite sum whose typical term is the product of a finite exponential sum (analogous to a Kloosterman sum) and a Bessel function of order k 1 . The main result is that, for any points z 1 and z 2 in the upper half-plane .~, the numbers m k-1 co,,(z 1, z2) (m= 1, 2, ...) are the Fourier coefficients of a modular form (in one variable) of weight k. More precisely, let D be the discriminant of K, e.=(D/ ) the character of K, and S(D, k, ~) the space of cusp forms of weight k for Fo(D ) with character e; then for fixed z 1, zzc .~, the function