2013
DOI: 10.1090/s0002-9939-2013-11840-x
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Divisibility of an eigenform by an eigenform

Abstract: Abstract. It has been shown in several settings that the product of two eigenforms is rarely an eigenform. In this paper we consider the more general question of when the product of an eigenform with any modular form is again an eigenform. We prove that this can occur only in very special situations. We then relate the divisibility of eigenforms to linear independence of vectors of Rankin-Selberg L-values.

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Cited by 4 publications
(11 citation statements)
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“…Proof of Theorem 3.1. If n = 0 this is multiplication, and so reduces to our previous result [2]. Otherwise we have that [E wt(f ) , S wt(g) ] n ⊆ S wt(f )+wt(g)+2n .…”
Section: Proofs Of Main Thoeremssupporting
confidence: 49%
See 2 more Smart Citations
“…Proof of Theorem 3.1. If n = 0 this is multiplication, and so reduces to our previous result [2]. Otherwise we have that [E wt(f ) , S wt(g) ] n ⊆ S wt(f )+wt(g)+2n .…”
Section: Proofs Of Main Thoeremssupporting
confidence: 49%
“…Proof of Theorem 3.2. If n = 0 the Rankin Cohen bracket operator is multiplication and so the theorem reduces to our previous result [2]. Assume n ≥ 1.…”
Section: Proofs Of Main Thoeremsmentioning
confidence: 70%
See 1 more Smart Citation
“…relations of the form h = f g where f, g, h are modular forms and f, h are eigenforms. The relation with Maeda's conjecture is discussed in Section 6 of [BJX11], and Theorem 1.5 implies the following result.…”
Section: Eigenforms Divisible By Eigenformsmentioning
confidence: 76%
“…Since the Hecke operators do not act on the entire algebra M of modular forms (they act differently on the graded pieces M k ), it seems reasonable that the one-dimensional coincidences are the only situation in which a product of eigenforms is an eigenform. Such questions have been studied by several authors, with the latest results appearing in a recent paper by Beyerl-James-Xue [BJX11]. They consider the more general question of divisibility of an eigenform by another eigenform, i.e.…”
Section: Eigenforms Divisible By Eigenformsmentioning
confidence: 99%