scholarly journals A critical case of Rallis inner product formula

2016 ◽  
Vol 60 (2) ◽  
pp. 201-222 ◽  
Author(s):  
ChenYan Wu
Keyword(s):  
Critical Case ◽  
Product Formula ◽  
Inner Product

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.

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

scholarly journals On the Inner Products of Some Deligne–Lusztig-type Representations

2018 ◽  
Vol 2020 (18) ◽  
pp. 5754-5773 ◽  
Author(s):  
Zhe Chen
Keyword(s):  
Product Formula ◽  
Inner Product ◽  
Reductive Groups ◽  
Inner Products ◽  
Discrete Valuation Rings ◽  
Valuation Rings

Abstract In this paper we introduce a family of Deligne–Lusztig-type varieties attached to connected reductive groups over quotients of discrete valuation rings, naturally generalising the higher Deligne–Lusztig varieties and some constructions related to the algebraisation problem raised by Lusztig. We establish the inner product formula between the representations associated to these varieties and the higherDeligne–Lusztig representations.

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