On Eisenstein series generated from twisting of the
geometric series
SHEN Li-chien
【摘 要】Abstract:In this paper,we will be dealing with the twisting of geometric series by the Dirichlet characters.In conjunction with the basic tool of Fourier transform,it can be used to generate all the Eisenstein series with respect to a family arithmetic groups. 【期刊名称】华东师范大学学报（自然科学版） 【年(卷),期】2017(000)006 【总页数】24
【关键词】Key words:Dirichlet character;conductor;Eisenstein series;Gaussian
sum;Kronecker
symbol;Mellin
transform;modular
form;Poisson summation formula;Weil’s converse theorem The paper is organized as follows.
The relevant properties of the Dirichlet characters and the geometric series twisted by the primitive characters are given in Sections 0 and 1.Sections 2 and 3 consist of the Mellin transform of the twisted geometric series,the derivation of the associated Fourier transform and a family of functions invariant under the Fourier transform.The Eisenstein series of weight k≥3 generated by the twisted Poisson summation formula are derived in Sections 4,5,6 and 7.In Sections 8 and 9,we provide the statement of Weil’s Converse Theorem as well as the
needed background materials and the proof of the modularity of the weight on Eisenstein with respect to certain arithmetic groups.In Section 10,we consider a twisted Weierstrass η function and examine its relevancy in the context of the late work of Kronecker on the theory of elliptic functions.
0 Basic properties of χ-twisted geometric series
Let N be a positive integer.Let χ be a Dirichlet character modulo N.Extend it to the set Z so that,for all integers m and n, (1)χ(1)=1; (2)χ(n+N)= χ(n); (3)χ(mn)= χ(m)χ(n); (4)χ(n)=0 if gcd(n,N)＞ 1.
Let N′be a positive integer which is divisible by N.For any character χ modulo N,we can form a character χ′modulo N′as follows:
We say that χ′is induced by the character χ.Let χ be a character modulo N.If there is a proper divisor d of N and a character modulo d which induces χ,then the character χ is said to be non-primitive,otherwise it is called primitive;and we say N is the conductor of χ if it is primitive modulo N.We note that if N is the conductor of a character,then either N=1 or N≥3.
We shall assume N≥3 unless indicated otherwise.
The Gaussian sum gk(χ)associated with the character χ is defined as
We denote g(χ)=g1(χ).We need a lemma(cf.[1,p.5]).
Lemma 0.1 Let χ be a primitive character modulo N.Then for any integer k,
We remark for gcd(k,N)=1,the above equality holds without χ being primitive;the primitiveness of χ is needed in showing that the left hand side of the identity is zero when gcd(k,N)＞1.
Let χ be a Dirichlet character.The series∑n χ(n)anis said to be the χ-twisted series of∑ nan. Consider
The function ? is generated by twisting the geometric serieswith the character χ: where|t|＜1.
We observe that,although ? is derived from the geometric series under the assumption:|t|＜1,it is() defined for all t∈C except possibly at the zeros of 1?tN. Let
χd(n)=denote
the is
Kronecker’s well-known
extension that
χdis
of
Jacobi primitive
symbol(cf.[2,p.35]).It
modulo|d|(cf.[3,p.347,Theorem 5]).For d=1,we define χ1(n)=1 for all n. We have
It is easy to verify the following lemma. Lemma 0.2 Let N be the conductor of χ.
由几何级数的扭曲生成的艾森斯坦级数
