Let $F$ be a function field over $\mathbb{F}_q$, $A$ its ring of regular functions outside a place $\infty$ and $\mathfrak{p}$ a prime ideal of $A$. At first, we develop Hida theory for Drinfeld modular forms of rank $r$ which are of slope zero for a suitably defined Hecke operator $\mathrm{U}_{\mathfrak{p}}$. Second, we show the existence in the finite slope case of families of Drinfeld modular forms varying continuously with the weight. Finally, we show a classicity result: an overconvergent Drinfeld modular form of sufficiently small slope with respect to the weight is a classical Drinfeld modular form.

中文翻译：

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Familles de formes modulaires de Drinfeld pour le groupe général linéaire

令 $F$ 是 $\mathbb{F}_q$ 上的函数域，$A$ 是 $\infty$ 外的正则函数环，$\mathfrak{p}$ 是 $A$ 的素理想。首先，我们为秩 $r$ 的 Drinfeld 模形式开发了 Hida 理论，对于适当定义的 Hecke 算子 $\mathrm{U}_{\mathfrak{p}}$ 斜率为零。其次，我们证明了在有限斜率情况下，Drinfeld 模形式族随权重连续变化的存在性。最后，我们展示了一个经典结果：相对于权重的斜率足够小的过收敛 Drinfeld 模形式是经典的 Drinfeld 模形式。