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].

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.

Remarks on the rightmost critical value of the triple product L-function

2021 ◽  
pp. 1-16
Author(s):  
Kengo Fukunaga ◽  
Kohta Gejima
Keyword(s):  
Critical Value ◽  
Product Formula ◽  
Orthogonal Basis ◽  
Triple Product ◽  
Explicit Formulas ◽  
Arithmetic Expression ◽  
Weighted Averages ◽  
Hecke Eigenforms ◽  
Hecke Eigenform ◽  
The Individual

Let [Formula: see text] be a normalized cuspidal Hecke eigenform. We give explicit formulas for weighted averages of the rightmost critical values of triple product [Formula: see text]-functions [Formula: see text], where [Formula: see text] and [Formula: see text] run over an orthogonal basis of [Formula: see text] consisting of normalized cuspidal Hecke eigenforms. Those explicit formulas provide us an arithmetic expression of the rightmost critical value of the individual triple product [Formula: see text]-functions.

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 regularized Siegel–Weil formula (the second term identity) and the Rallis inner product formula

2014 ◽  
Vol 198 (3) ◽  
pp. 739-831 ◽  
Author(s):  
Wee Teck Gan ◽  
Yannan Qiu ◽  
Shuichiro Takeda
Keyword(s):  
Product Formula ◽  
Inner Product

scholarly journals Inner product formula for Yoshida lifts

2017 ◽  
Vol 42 (2) ◽  
pp. 215-253
Author(s):  
Ming-Lun Hsieh ◽  
Kenichi Namikawa
Keyword(s):  
Product Formula ◽  
Inner Product
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