Lifting of Modular Forms

Jitendra Bajpai

doi:10.5802/pmb.27

Jitendra Bajpai ¹

¹ Mathematisches Institut, Georg-August Universität Göttingen, D-37073 Germany.

Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2019), pp. 5-20.

Abstract
Résumé

The existence and construction of vector-valued modular forms (vvmf) for any arbitrary Fuchsian group $G$ , for any representation $ρ : G ⟶ {GL}_{d} (ℂ)$ of finite image can be established by lifting scalar-valued modular forms of the finite index subgroup $ker (ρ)$ of $G$ . In this article vvmf are explicitly constructed for any admissible multiplier (representation) $ρ$ , see Section 3 for the definition of admissible multiplier. In other words, the following question has been partially answered: For which representations $ρ$ of a given $G$ , is there a vvmf with at least one nonzero component?

L’existence et construction de formes modulaires vectorielles (vvmf) pour un groupe Fuchsien arbitraire $G$ et pour une représentation $ρ : G ⟶ {GL}_{d} (ℂ)$ d’image finie peut être établie en relevant des formes modulaires scalaires pour le sous-groupe d’indice fini $ker (ρ)$ de $G$ . Dans cet article, des vvmf sont explicitement construites pour tout multiplicateur admissible (représentation) $ρ$ (voir paragraphe 3 pour la définition du multiplicateur admissible). En d’autres termes, on a partiellement répondu à la question suivante : Pour quelles représentations $ρ$ d’un groupe $G$ donné, existe-t-il une vvmf avec au moins une composante non nulle ?

Received: 2017-06-02
Published online: 2019-10-15

DOI: 10.5802/pmb.27

Classification: 11F03, 11F55, 30F35
Keywords: Fuchsian group, Vector-valued modular form, Induced representation

Author's affiliations:

Jitendra Bajpai ¹

¹ Mathematisches Institut, Georg-August Universität Göttingen, D-37073 Germany.

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Jitendra Bajpai. Lifting of Modular Forms. Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2019), pp. 5-20. doi : 10.5802/pmb.27. https://pmb.centre-mersenne.org/articles/10.5802/pmb.27/

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