Embed Size px x x x x Abstract The specic aims of this paper are to dene a Jacobi-Eisenstein series of weight two on congruence. Jacobi subgroup and to compute its Fourier expansion coecients in detail. To overcome the diculties that.

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Embed Size px x x x x Abstract The specic aims of this paper are to dene a Jacobi-Eisenstein series of weight two on congruence. Jacobi subgroup and to compute its Fourier expansion coecients in detail. To overcome the diculties that. A Jacobi-Eisenstein series of weight two on congruence Jacobi subgroup. Sci China. Eisenstein series play a crucial role in the theory of modular forms and number theory.

They are usednot only to construct examples of modular forms, but also to obtain the number of representations of anatural number n as a sum of squares. It is very interesting that the coecients of the Fourier expansionof Eisenstein series of half weight have a close relation with the class number relation which has adeep number theory and geometry background.

Hirzebruch, Zagier [4], Cohen [2], et al. Choie[1] dened a Jacobi-Eisenstein series on full Jacobi group by Heckes trick and studied correspondencesamong many Eisenstein series. However, as the same as in the elliptic modular form case, there is nononzero holomorphic Jacobi form of weight two which is invariant under the action of full Jacobi group.

But for the congruence Jacobi group, for example 0 4 J , there exist nonzero holomorphic Jacobi formsof weight two. In this paper we will dene a non-holomorphic Jacobi-Eisenstein series of weight two oncongruence Jacobi group 0 4 J , and express its Fourier coecients by class number relation.

This isalso the rst step to construct nonzero holomorphic Jacobi forms of weight two on congruence Jacobigroup 0 4 J. In Section 2, we will review the notations and denition of Jacobi forms, then give the denition ofJacobi-Eisenstein series on congruence Jacobi group and state our two main theorems.

In Sections 3and 4, we give the proofs of the two main theorems respectively. Now we x some notations in this paper. For integers a, b, sometimes, we use the symbol a, b instead of gcd a, b , to denote the greatest commondivisor of a and b. For z. In this section we recall the denition of Jacobi forms and dene the Jacobi-Eisenstein series of weighttwo and index one on congruence Jacobi subgroup.

Let H be the complex upper half-plane, and C the complex plane, respectively. Let k,m be positive integers, and SL 2,Z be a subgroup of finite index. A Jacobiform of weight k and index m on a subgroup is a holomorphic function : H C C satisfying :.

For more details about Jacobi forms, we refer to [3]. A Jacobi-Eisenstein series of weight k and index m, is dened by the following:. To overcome this trouble, we use thewell-known Heckes convergence trick by dening. Before stating main theorems, we recall some facts about the quadratic eld. Let 0 be the discriminantof the quadratic eld Q. In the following, let mp be the number such that pmpf. Theorem 1. Let the notations be defined as above. The next theorem will give the Fourier expansion ofE2,1;4 , z more explicitly.

Theorem 2. Let E2,1;4 , z be defined as above. Then E2,1;4 , z has the Fourier expansion as follows:. We do not repeated here. We split the sum over c, d as twoparts, according to c being 0 or not.

The function P2,1 , z; s is periodic in z as well as the real part u of , so it has a Fourier expansionin z and u. See Full Reader. View Download 1 Category Documents.

September Vol. To overcome the diculties that the Jacobi-Eisenstein series of weight two is not convergent absolutely, we use the Heckes trick. Sci China Math, , 53 9 : , doi: Corresponding author LU HongWen et al. Sci China Math September Vol. We denote by LU HongWen et al. To compute the summation n,r s , we follow Eichler and Zagiers trick [3].

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