Products of vector valued Eisenstein series

Raum, Martin

doi:10.1515/forum-2014-0198

Cited by 17 publications

(91 citation statements)

References 26 publications

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“…Without loss of generality, we can assume that l ≤ l . In complete analogy with [23], we note that the right-hand side of (4.4) is a T f -module. It therefore suffices to show the following: Any scalar multiple of a newform f for the congruence subgroup Γ 0 (N ) vanishes, if for t = (l +l −k)/ 2 ∈ Z ≥0 and all D, the (holomorphic) modular form f D is orthogonal to the almost holomorphic modular form…”

Section: Hecke Operators At All Places In Genusmentioning

confidence: 99%

“…If v = p is a prime, then T p generalizes classical Hecke operators for Siegel modular forms. It extends the vectorvalued Hecke operators for elliptic modular forms that we have already defined in [23]. The polynomial algebra T generated by formal elements T v acts by means of these Hecke operators on the hyper-algebra of Siegel modular forms.…”

Section: Introductionmentioning

confidence: 94%

“…Types can, however, be arbitrary dimensional. We adopt an example from [23], which illustrates how to compute products in M (g) (• ⊗ •) of modular forms with nontrivial type. For convenience, we give Sage code [16] 1 , which the reader can modify to perform his or her own computation -the code is not optimized for performance, but for clarity.…”

Section: Computing the Hyper-algebra Product In Sagementioning

confidence: 99%

“…The hyper-algebra structure and the action of the formal Hecke algebra make it rather easy to formulate essential surjectivity for products of modular forms. The blueprint for such results are the ones in [6,23]. The first of them, phrased in the language that we have developed reads as follows.…”

Section: Essential Surjectivity Of Tensor Products Of Modular Formsmentioning

confidence: 99%

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Hyper-Algebras of Vector-Valued Modular Forms

Raum¹

2018

SIGMA

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We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of Q, acting on these hyper-algebras. These definitions bridge the classical and representation theoretic approach to Siegel modular forms. Combining both the product structure and the action of Hecke operators, we prove in the case of elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from products involving only two fixed Eisenstein series. As a byproduct, we obtain inclusions of cuspidal automorphic representations into the tensor product of global principal series.

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Section: Hecke Operators At All Places In Genusmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 94%

Section: Computing the Hyper-algebra Product In Sagementioning

confidence: 99%

Section: Essential Surjectivity Of Tensor Products Of Modular Formsmentioning

confidence: 99%

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Hyper-Algebras of Vector-Valued Modular Forms

Raum¹

2018

SIGMA

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“…Recall the induction map from modular forms for a subgroup Γ ⊆ Γ (2) to modular forms of type ρ Γ , which, for example, was spelled out in [WR14]. The image of…”

Section: Acknowledgmentmentioning

confidence: 99%

Harmonic Weak Siegel–Maaß Forms I

Raum

2016

Int Math Res Notices

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Given a non-holomorphic SaitoKurokawa lift we construct a preimage under the vector-valued lowering operator. In analogy with the case of harmonic weak elliptic Maaß forms, this preimage allows for a natural decomposition into a meromorphic and a non-holomorphic part. In this way every harmonic weak Siegel Maaß form gives rise to a Siegel mock modular form. Siegel mock modular forms Harish-Chandra and (g, K)-modules Holomorphic parts of Fourier expansions MSC Primary 11F37 MSC Secondary 11F46, 11F70, 22E50M ORE than 10 years ago Zwegers [Zwe02], and Bruinier and Funke [BF04] independently found definitions of harmonic weak Maaß forms. Zwegers focused on mock theta functions whose modular properties had been a long standing miracle since Ramanujan came up with them in 1920 in his death bed letter. Bruinier and Funke were inspired by the Kudla program and they used harmonic weak Maaß forms to describe dualities between theta lifts. Both approaches to harmonic weak Maaß forms together produced a rich and novel research area, leading to the resolution of vital conjectures in combinatorics, topology, and many other fields-see, for example, [Ono09] for an overview and [BO06; BO10a; DIT11; DMZ12] for some of the applications. In this paper, we suggest an extension of the concept of harmonic weak Maaß forms to the case of Siegel modular forms of genus 2, by building up on the ideas of [BF04].In order to state some defining formulas, we let τ = x + i y ∈ H be a variable in the Poincaré upper half plane and 0 ≤ k an even integer. Analytic aspects of the theory of harmonic weak Maaß forms have been dominated by two facts. First, the weight-k hyperbolic Laplace operator ∆ k can be written as a composi-Second, the ξ-operator gives rise to a short exact sequence of the space ! M k of weakly holomorphic modular forms, the space S 2−k of cusp forms, and the space S k of harmonic weak Maaß forms whose image under ξ k is a cusp form.This sequence puts the operator ξ k into the center of the theory. One can define harmonic weak Maaß forms as the modular preimages of weakly holomorphic modular forms of weight 2 − k under ξ k . Note that the ξ-operator is essentially the Maaß lowering operator L. Concretely, we haveHarmonic weak Siegel Maaß forms that we study in this paper are vector-valued. Recall that every complex representation of GL 2 (C) can be viewed as a weight for genus 2 Siegel modular forms. The classical case of weight k modular forms corresponds to the representation detIn our case of genus 2 Siegel modular forms, every complex, irreducible representation of GL 2 (C) can be written as a (tensor) product det k sym l where sym l is the l -th symmetric power representation of GL 2 (C). It is necessary to introduce vector-valued Siegel modular forms, because the above target space S 2−k of ξ k will be replaced by SK S (2) det −k/2 sym k : space of non-holomorphic Saito-Kurokawa lifts of weight k cusp forms.We revisit the non-holomorphic Saito-Kurokawa lift in Section 1. Note that non-holomorphic Saito-Kurokawa li...

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Hecke relations in rational conformal field theory

Harvey

2018

J. High Energ. Phys.

108

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We define Hecke operators on vector-valued modular forms of the type that appear as characters of rational conformal field theories (RCFTs). These operators extend the previously studied Galois symmetry of the modular representation and fusion algebra of RCFTs to a relation between RCFT characters. We apply our results to derive a number of relations between characters of known RCFTs with different central charges and also explore the relation between Hecke operators and RCFT characters as solutions to modular linear differential equations. We show that Hecke operators can be used to construct an infinite set of possible characters for RCFTs with two independent characters and increasing central charge. These characters have multiplicity one for the vacuum representation, positive integer coefficients in their q expansions, and are associated to a two-dimensional representation of the modular group which leads to non-negative integer fusion coefficients as determined by the Verlinde formula.

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Products of vector valued Eisenstein series

Cited by 17 publications

References 26 publications

Hyper-Algebras of Vector-Valued Modular Forms

Hyper-Algebras of Vector-Valued Modular Forms

Harmonic Weak Siegel–Maaß Forms I

Hecke relations in rational conformal field theory

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