Lifting of Modular Forms
Publications Mathématiques de Besançon no. 1  (2019), p. 5-20

The existence and construction of vector-valued modular forms (vvmf) for any arbitrary Fuchsian group $\mathrm{G}$, for any representation $\rho :\mathrm{G}⟶{\mathrm{GL}}_{d}\left(ℂ\right)$ of finite image can be established by lifting scalar-valued modular forms of the finite index subgroup $ker\left(\rho \right)$ of $\mathrm{G}$. In this article vvmf are explicitly constructed for any admissible multiplier (representation) $\rho$, see Section 3 for the definition of admissible multiplier. In other words, the following question has been partially answered: For which representations $\rho$ of a given $\mathrm{G}$, 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 $\rho :\mathrm{G}⟶{\mathrm{GL}}_{d}\left(ℂ\right)$ d’image finie peut être établie en relevant des formes modulaires scalaires pour le sous-groupe d’indice fini $ker\left(\rho \right)$ de $G$. Dans cet article, des vvmf sont explicitement construites pour tout multiplicateur admissible (représentation) $\rho$ (voir paragraphe 3 pour la définition du multiplicateur admissible). En d’autres termes, on a partiellement répondu à la question suivante : Pour quelles représentations $\rho$ d’un groupe $G$ donné, existe-t-il une vvmf avec au moins une composante non nulle ?

Published online : 2019-10-15
Classification:  11F03,  11F55,  30F35
Keywords: Fuchsian group, Vector-valued modular form, Induced representation
@article{PMB_2019___1_5_0,
author = {Jitendra Bajpai},
title = {Lifting of Modular Forms},
journal = {Publications Math\'ematiques de Besan\c con},
publisher = {Presses universitaires de Franche-Comt\'e},
number = {1},
year = {2019},
pages = {5-20},
language = {en},
url = {https://pmb.centre-mersenne.org/item/PMB_2019___1_5_0}
}

Bajpai, Jitendra. Lifting of Modular Forms. Publications Mathématiques de Besançon, no. 1 (2019), pp. 5-20. pmb.centre-mersenne.org/item/PMB_2019___1_5_0/

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