Let $p$ be a rational prime and $q$ a power of $p$. Let $\wp$ be a monic irreducible polynomial of degree $d$ in ${\mathbb{F}}_q[t]$. In this paper, we define an analogue of the Hodge–Tate map which is suitable for the study of Drinfeld modules over ${\mathbb{F}}_q[t]$ and, using it, develop a geometric theory of $\wp$‐adic Drinfeld modular forms similar to Katz's theory in the case of elliptic modular forms. In particular, we show that for Drinfeld modular forms with congruent Fourier coefficients at $\infty$ modulo ${\wp }^n$, their weights are also congruent modulo $(q^d-1)p^{\lceil {\log }_p(n)\rceil }$, and that Drinfeld modular forms of level ${\mathrm{\Gamma }}_1(\mathfrak{n})\cap {\mathrm{\Gamma }}_0(\wp )$, weight $k$ and type $m$ are $\wp$‐adic Drinfeld modular forms for any tame level $\mathfrak{n}$ with a prime factor of degree prime to $q-1$.

中文翻译：

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Drinfeld模块的对偶性和Drinfeld模块化形式的偶数特性

让 $p$ 成为理性的素数 $q$ 的力量 $p$。让$\wp$ 是单数不可约的多项式 $d$ 在 ${\mathbb{F}}_q[Ť]$。在本文中，我们定义了Hodge-Tate地图的类似物，适用于研究Drinfeld模块${\mathbb{F}}_q[Ť]$ 并利用它发展出一个几何理论 $\wp$-adin Drinfeld模块化形式类似于椭圆模块化形式的Katz理论。特别是，我们证明了对于具有一致傅立叶系数的Drinfeld模块化形式，$\infty$ 模数 ${\wp }^{ñ}$，它们的权重也是全模的 $（q^d-\mathrm{1个}）p^{\lceil {\mathrm{日志}}_p（ñ）\rceil }$，以及Drinfeld级别的模块化形式 ${\mathrm{\Gamma }}_{\mathrm{1个}}（\mathfrak{ñ}）\cap {\mathrm{\Gamma }}_0（\wp ）$，重量 $ķ$ 和类型 $米$ 是 $\wp$-adic Drinfeld模块化形式，适用于任何温顺等级 $\mathfrak{ñ}$ 具有素数的素数 $q-\mathrm{1个}$。