On certain Drinfeld modular forms of higher rank

Dirk Basson; Florian Breuer

doi:10.5802/jtnb.1003

Dirk Basson ; Florian Breuer

Journal de Théorie des Nombres de Bordeaux, Tome 29 (2017) no. 3, pp. 827-843.

Résumé
Abstract

Nous donnons une introduction aux formes modulaires de Drinfeld pour des sous-groupes de congruence principaux de ${GL}_{r} (𝔽_{q} [t])$ , et puis nous construisons un analogue en rang $r$ de la fonction $h$ . Nous montrons que cette fonction est cuspidale de poids $(q^{r} - 1) / (q - 1)$ et de type 1 et qu’elle satisfait une formule de produit. Dans ce but, nous calculons le développement à l’infini des séries d’Eisenstein de poids 1 et de nivaux $N \in 𝔽_{q} [t]$ .

We give an introduction to Drinfeld modular forms for principal congruence subgroups of ${GL}_{r} (𝔽_{q} [t])$ , and then construct a rank $r$ analogue of the $h$ -function. We show that this function is a cusp form of weight $(q^{r} - 1) / (q - 1)$ and type 1 which satisfies a product formula. Along the way, we compute the expansion at infinity of weight one Eisenstein series of level $N \in 𝔽_{q} [t]$ .

Reçu le : 2016-09-11
Révisé le : 2017-06-07
Accepté le : 2017-06-15
Publié le : 2017-12-12

DOI : https://doi.org/10.5802/jtnb.1003
Classification : 11F52, 11G09
Mots clés : Drinfeld modular forms, Drinfeld modules

@article{JTNB_2017__29_3_827_0,
     author = {Dirk Basson and Florian Breuer},
     title = {On certain {Drinfeld} modular forms of higher rank},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {827--843},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
     number = {3},
     year = {2017},
     doi = {10.5802/jtnb.1003},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1003/}
}

Dirk Basson; Florian Breuer. On certain Drinfeld modular forms of higher rank. Journal de Théorie des Nombres de Bordeaux, Tome 29 (2017) no. 3, pp. 827-843. doi : 10.5802/jtnb.1003. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1003/

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