On certain Drinfeld modular forms of higher rank

Basson, Dirk; Breuer, Florian

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

Nous donnons une introduction aux formes modulaires de Drinfeld pour des sous-groupes de congruence principaux de ${\mathrm{GL}}_r\left({𝔽}_q\left[t\right]\right)$, 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 $N\in {𝔽}_q\left[t\right]$.

We give an introduction to Drinfeld modular forms for principal congruence subgroups of ${\mathrm{GL}}_r\left({𝔽}_q\left[t\right]\right)$, 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 $N\in {𝔽}_q\left[t\right]$.

Classification : 11F52, 11G09
Mots clés : Drinfeld modular forms, Drinfeld modules

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