scholarly journals RANKIN’S METHOD AND JACOBI FORMS OF SEVERAL VARIABLES

2010 ◽  
Vol 88 (1) ◽  
pp. 131-143 ◽  
Author(s):  
B. RAMAKRISHNAN ◽  
BRUNDABAN SAHU
Keyword(s):  
Modular Form ◽  
Eisenstein Series ◽  
Cusp Form ◽  
Inner Product ◽  
Jacobi Forms ◽  
Several Variables ◽  
Special Value

AbstractFollowing R. A. Rankin’s method, D. Zagier computed the nth Rankin–Cohen bracket of a modular form g of weight k1 with the Eisenstein series of weight k2, computed the inner product of this Rankin–Cohen bracket with a cusp form f of weight k=k1+k2+2n and showed that this inner product gives, up to a constant, the special value of the Rankin–Selberg convolution of f and g. This result was generalized to Jacobi forms of degree 1 by Y. Choie and W. Kohnen. In this paper, we generalize this result to Jacobi forms defined over ℋ×ℂ(g,1).

scholarly journals The inner product of an automorphic wave form with the pullback of an Eisenstein series

1987 ◽  
Vol 108 ◽  
pp. 93-119
Author(s):  
Shinji Niwa
Keyword(s):  
Eisenstein Series ◽  
Automorphic Forms ◽  
Theta Functions ◽  
Inner Product ◽  
Wave Form ◽  
Prove Theorem ◽  
Wave Forms ◽  
Special Value ◽  
Special Case ◽  
Hilbert Type

In this paper we shall show a relation between a special value of an automorphic wave form and the inner product of the automorphic wave form with the pullback of an Eisenstein series on the upper half space. The main theorem is Theorem 3 in the end of this paper. As is shown in P. B. Garrett [13], pullbacks of Eisenstein series on Siegel upper half spaces have interesting properties as a kernel function of an integral operator. It is natural to try to investigate pullbacks of Eisenstein series of Hilbert type. We can say that Theorem 3 clarifies a property of such pullbacks in a special case. The idea of the proof is a lifting of automorphic forms by theta functions. We discuss a lifting of automorphic wave forms in 1, 2 and 3, and obtain Theorem 2 in the end of 3 as a result. We can prove Theorem 3 without much difficulty by using Theorem 2.

Lambert series associated to Hilbert modular form

2021 ◽  
pp. 1-15
Author(s):  
Rishabh Agnihotri
Keyword(s):  
Asymptotic Expansion ◽  
Modular Form ◽  
Cusp Form ◽  
Riemann Zeta Function ◽  
Hilbert Modular Form ◽  
Lambert Series ◽  
Nontrivial Zeros ◽  
Hilbert Modular ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form [Formula: see text] should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion.

THE MINIMAL MODULAR FORM ON QUATERNIONIC

2020 ◽  
pp. 1-34
Author(s):  
Aaron Pollack
Keyword(s):  
Modular Form ◽  
Eisenstein Series ◽  
Explicit Form ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Reductive Group ◽  
Automorphic Function ◽  
Unipotent Radical ◽  
Exceptional Groups ◽  
Dynkin Type

Suppose that $G$ is a simple reductive group over $\mathbf{Q}$ , with an exceptional Dynkin type and with $G(\mathbf{R})$ quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\unicode[STIX]{x1D703}_{Gan}$ on quaternionic $E_{8}$ and some applications. The $Sym^{8}(V_{2})$ -valued automorphic function $\unicode[STIX]{x1D703}_{Gan}$ is a weight 4, level one modular form on $E_{8}$ , which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic $E_{7},E_{6}$ and $G_{2}$ . We also discuss a family of degenerate Heisenberg Eisenstein series on the groups $G$ , which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups $\operatorname{GSp}_{2n}$ .

scholarly journals All modular forms of weight 2 can be expressed by Eisenstein series

2020 ◽  
Vol 6 (3) ◽  
Author(s):  
Martin Raum ◽  
Jiacheng Xia
Keyword(s):  
Modular Form ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Linear Combinations ◽  
Integral Weight ◽  
Elliptic Modular Form

Abstract We show that every elliptic modular form of integral weight greater than 1 can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central $$\mathrm{L}$$ L -values present in all previous work. For weights greater than 2, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.

Polynomial identities between Hecke eigenforms

2019 ◽  
Vol 15 (10) ◽  
pp. 2135-2150
Author(s):  
Dianbin Bao
Keyword(s):  
Eisenstein Series ◽  
Product Formula ◽  
Inner Product ◽  
Polynomial Identities ◽  
Number Of Solutions ◽  
Hecke Eigenforms

In this paper, we study solutions to [Formula: see text], where [Formula: see text] are Hecke newforms with respect to [Formula: see text] of weight [Formula: see text] and [Formula: see text]. We show that the number of solutions is finite for all [Formula: see text]. Assuming Maeda’s conjecture, we prove that the Petersson inner product [Formula: see text] is nonzero, where [Formula: see text] and [Formula: see text] are any nonzero cusp eigenforms for [Formula: see text] of weight [Formula: see text] and [Formula: see text], respectively. As a corollary, we obtain that, assuming Maeda’s conjecture, identities between cusp eigenforms for [Formula: see text] of the form [Formula: see text] all are forced by dimension considerations. We also give a proof using polynomial identities between eigenforms that the [Formula: see text]-function is algebraic on zeros of Eisenstein series of weight [Formula: see text].

ANALYTIC CONTINUATION OF NONANALYTIC VECTOR-VALUED EISENSTEIN SERIES

2009 ◽  
Vol 05 (05) ◽  
pp. 805-830
Author(s):  
KAREN TAYLOR
Keyword(s):  
Analytic Continuation ◽  
Eisenstein Series ◽  
Complex Plane ◽  
Cusp Form ◽  
Vector Valued

In this paper, we introduce a vector-valued nonanalytic Eisenstein series appearing naturally in the Rankin–Selberg convolution of a vector-valued modular cusp form associated to a monomial representation ρ of SL(2,ℤ). This vector-valued Eisenstein series transforms under a representation χρ associated to ρ. We use a method of Selberg to obtain an analytic continuation of this vector-valued nonanalytic Eisenstein series to the whole complex plane.

scholarly journals Differential Operators on Jacobi Forms of Several Variables

2000 ◽  
Vol 82 (1) ◽  
pp. 140-163 ◽  
Author(s):  
YoungJu Choie ◽  
Haesuk Kim
Keyword(s):  
Differential Operators ◽  
Jacobi Forms ◽  
Several Variables

Rankin–Cohen brackets on Jacobi forms of several variables and special values of certain Dirichlet series

2019 ◽  
Vol 15 (05) ◽  
pp. 925-933
Author(s):  
Abhash Kumar Jha ◽  
Brundaban Sahu
Keyword(s):  
Dirichlet Series ◽  
Scalar Product ◽  
Fourier Coefficients ◽  
Differential Operators ◽  
Cusp Forms ◽  
Jacobi Forms ◽  
Linear Map ◽  
Special Values ◽  
Several Variables

We construct certain Jacobi cusp forms of several variables by computing the adjoint of linear map constructed using Rankin–Cohen-type differential operators with respect to the Petersson scalar product. We express the Fourier coefficients of the Jacobi cusp forms constructed, in terms of special values of the shifted convolution of Dirichlet series of Rankin–Selberg type. This is a generalization of an earlier work of the authors on Jacobi forms to the case of Jacobi forms of several variables.

A p-ADIC LIMIT OF SIEGEL–EISENSTEIN SERIES OF PRIME LEVEL q

2008 ◽  
Vol 04 (05) ◽  
pp. 735-746 ◽  
Author(s):  
YOSHINORI MIZUNO
Keyword(s):  
Modular Form ◽  
Eisenstein Series ◽  
Simple Formula ◽  
Fourier Coefficients ◽  
Siegel Modular Form

We show that a p-adic limit of a Siegel–Eisenstein series of prime level q becomes a Siegel modular form of level pq. This paper contains a simple formula for Fourier coefficients of a Siegel–Eisenstein series of degree two and prime levels.

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