Un théorème de transcendance en caractéristique finie

Dion, Sophie

Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 57-82.

Soit $p$ un nombre premier et $q=p^r$, où $r\in \mathbb{N}$ est non nul. On considère le corps $k={𝔽}_q\left(T\right)$, dont on note $\overline{k}$ une clôture algébrique. On peut associer à tout module de Drinfeld de rang $2$, défini sur $\overline{k}$, un invariant modulaire $j$, ainsi que des fonctions modulaires. On s'intéresse à l'une d'entre elles, notée $\overline{\Delta }$. On montre en particulier, que si $\left|t\right|<q^{-\frac{1}{q-1}}$ alors l'un au moins des deux nombres $j\left(t\right)$ et $\overline{\Delta }\left(t\right)$ est transcendant sur $k$.

Let $p$ be a prime number, and $q=p^r$, with $r\in \mathbb{N}\setminus \left\{0\right\}$. Let $k={𝔽}_q\left(T\right)$, and $\overline{k}$ be its algebraic closure. We can associate to any Drinfeld module of rank $2$, a modular invariant $j$, and also modular functions. We are interesting in one of them, denoted $\overline{\Delta }$. In particular, we show that if $\left|t\right|<q^{-\frac{1}{q-1}}$ then at least one of the two numbers $j\left(t\right)$ and $\overline{\Delta }\left(t\right)$ is transcendental over $k$.

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