On Drinfeld modular forms of higher rank

Ernst-Ulrich Gekeler

doi:10.5802/jtnb.1005

Ernst-Ulrich Gekeler ¹

¹ FR Mathematik Universität des Saarlandes Campus E2 4 66123 Saarbrücken, Germany

Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 3, pp. 875-902.

Abstract
Résumé

We study Drinfeld modular forms for the modular group $Γ = GL (r, 𝔽_{q} [T])$ on the Drinfeld symmetric space $Ω^{r}$ , where $r \geq 2$ . Results include the existence of a $(q - 1)$ -th root (up to constants) $h$ of the discriminant function $Δ$ , the description of the growth/decay of the standard forms $g_{1}, g_{2}, \dots g_{r - 1}$ , $Δ$ on the fundamental domain $ℱ$ of $Γ$ , and the reduction of these forms on the central part $ℱ_{o}$ of $ℱ$ . The results are exemplified in detail for $r = 3$ .

Nous étudions les formes modulaires pour le groupe $Γ = GL (r, 𝔽_{q} [T])$ sur l’espace symétrique $Ω^{r}$ de Drinfeld, où $r \geq 2$ . Parmi nos résultats, on a l’existence d’une racine $(q - 1)$ -ième (à une constante près) $h$ de la fonction discriminant $Δ$ , la description de la (dé-)croissance des formes élémentaires $g_{1}, g_{2}, \dots, g_{r - 1}, Δ$ dans le domaine fondamental $ℱ$ de $Γ$ , et la réduction de ces formes sur la partie centrale $ℱ_{o}$ de $ℱ$ . Nous étudions avec plus de détail le cas de $r = 3$ .

Received: 2016-06-12
Revised: 2017-01-08
Accepted: 2017-02-02
Published online: 2017-12-12

DOI: 10.5802/jtnb.1005

Classification: 11F52, 11G09, 14G22
Mots-clés : Drinfeld modular forms, Drinfeld discriminant function; Bruhat–Tits building

Author's affiliations:

Ernst-Ulrich Gekeler ¹

¹ FR Mathematik Universität des Saarlandes Campus E2 4 66123 Saarbrücken, Germany

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {On {Drinfeld} modular forms of higher rank},
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Ernst-Ulrich Gekeler. On Drinfeld modular forms of higher rank. Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 3, pp. 875-902. doi : 10.5802/jtnb.1005. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1005/

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