Sturm’s operator for scalar weight in arbitrary genus

Maurischat, Kathrin

doi:10.1142/s1793042117501482

Cited by 2 publications

(4 citation statements)

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“…However, in case of weight κ = 3 and rank m = 2 we showed jointly with R. Weissauer ( [8]) that Sturm's operator produces, along with the holomorphic projection, a second term ph(f + . Later ( [7]) we generalized this result to general rank m > 2 and κ = m + 1. However, the phenomenon of arising phantom terms in case of small weight is rather non-understood.…”

Section: Introductionmentioning

confidence: 75%

“…However, in case of weight

κ = 3

and rank

m = 2

we showed jointly with R. Weissauer ([8]) that Sturm's operator produces, along with the holomorphic projection, a second term

p h false(f false) \in {[Γ, κ]}_{0}

{S t}_{κ} false(f false) = {p r}_{h o l} false(f false) + p h false(f false) .

This phantom term

p h false(f false) = {S t}_{κ} ({normalΔ}_{+}^{[m]} false(h false))

arises as the non‐holomorphic Maass shift of a holomorphic form

h \in [Γ, κ - 2]

of weight one (see section 4 for the exact definition of

Δ_{+}^{false[m false]}

. Later ([7]) we generalized this result to general rank

m > 2

and

κ = m + 1

. However, the phenomenon of arising phantom terms in case of small weight is rather non‐understood.…”

Section: Introductionmentioning

confidence: 99%

“…Our results obtained so far are limited by the explicit computability of phantom terms. Nevertheless, by [7, 8], and the above, the following interpretation is at hand. A holomorphic cusp form of weight ρ generates a holomorphic representation of the symplectic group G of minimal K ‐type ρ.…”

Section: Introductionmentioning

confidence: 99%

“…The wall orthogonal to all short simple roots is given by

(r - 1, r - 2, \dots, r - m)

for

r \geq m + 1

. Here, [7] suggests that Sturm's operator realizes the holomorphic projection operator as long as

r > m + 1

, i.e. apart from the apex δ of the cone belonging to the minimal K ‐type

(m + 1, \dots, m + 1)

.…”

Section: Introductionmentioning

confidence: 99%

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Sturm's operator acting on vector valued K‐types

Maurischat

2020

Mathematische Nachrichten

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We define Sturm's operator for vector valued Siegel modular forms obtaining an explicit description of their holomorphic projection in case of large absolute weight.However, for small absolute weight, Sturm's operator produces phantom terms in addition. This confirms our earlier results for scalar Siegel modular forms. K E Y W O R D Sholomorphic projection, integral operator, Siegel modular forms, sturm operator M S C ( 2 0 1 0 ) 11F30, 11F46This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

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Section: Introductionmentioning

confidence: 75%

“…However, in case of weight