scholarly journals Rationality of the Petersson Inner Product of Cohen’s Kernels

2021 ◽  
pp. 1-19
Author(s):  
Yuanyi You ◽  
Yichao Zhang
Keyword(s):  
Eisenstein Series ◽  
Fourier Expansion ◽  
Inner Product ◽  
Double Eisenstein Series

By explicitly calculating and then analytically continuing the Fourier expansion of the twisted double Eisenstein series [Formula: see text] of Diamantis and O’Sullivan, we prove a formula of the Petersson inner product of Cohen’s kernel and one of its twists, and obtain a rationality result. This extends a result of Kohnen and Zagier.

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 $\operatorname{GSp}_{2n}$ .

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].

scholarly journals Dirichlet series in the theory of Siegel modular forms

1984 ◽  
Vol 95 ◽  
pp. 73-84 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Dirichlet Series ◽  
Eisenstein Series ◽  
Half Space ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Siegel Modular Forms

We are concerned with Dirichlet series which appear in the Fourier expansion of the non-analytic Eisenstein series on the Siegel upper half space Hm of degree m. In the case of m = 2 Kaufhold [1] evaluated them. Here we treat the general cases by a different method.

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).

SPECIAL VALUES OF L-FUNCTIONS ON GSp4 × GL2 AND THE NON-VANISHING OF SELMER GROUPS

2010 ◽  
Vol 06 (08) ◽  
pp. 1901-1926 ◽  
Author(s):  
JIM BROWN
Keyword(s):  
Eisenstein Series ◽  
Product Formula ◽  
Inner Product ◽  
Selmer Groups ◽  
Selmer Group ◽  
Special Values ◽  
New Form

In this paper, we show how one can use an inner product formula of Heim giving the inner product of the pullback of an Eisenstein series from Sp10 to Sp 2 × Sp 4 × Sp 4 with a new-form on GL2 and a Saito–Kurokawa lift to produce congruences between Saito–Kurokawa lifts and non-CAP forms. This congruence is in part controlled by the L-function on GSp 4 × GL 2. The congruence is then used to produce nontrivial torsion elements in an appropriate Selmer group, providing evidence for the Bloch–Kato conjecture.

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