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Ernst-Ulrich Gekeler
On Drinfeld modular forms of higher rank
Journal de théorie des nombres de Bordeaux, 29 no. 3 (2017), p. 875-902
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Class. Math.: 11F52, 11G09, 14G22
Keywords: Drinfeld modular forms, Drinfeld discriminant function; Bruhat–Tits building

Résumé - Abstract

We study Drinfeld modular forms for the modular group $\Gamma = \mathrm{GL}(r,\mathbb{F}_q[T])$ on the Drinfeld symmetric space $\Omega ^r$, where $r \ge 2$. Results include the existence of a $(q-1)$-th root (up to constants) $h$ of the discriminant function $\Delta $, the description of the growth/decay of the standard forms $g_1,g_2,\dots g_{r-1}$, $\Delta $ on the fundamental domain $\mathcal{F}$ of $\Gamma $, and the reduction of these forms on the central part $\mathcal{F}_{o}$ of $\mathcal{F}$. The results are exemplified in detail for $r = 3$.

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