We develop tools for constructing rigid analytic trivializations for Drinfeld modules as infinite products of Frobenius twists of matrices, from which we recover the rigid analytic trivialization given by Pellarin in terms of Anderson generating functions. One advantage is that these infinite products can be obtained from only a finite amount of initial calculation, and consequently we obtain new formulas for periods and quasi-periods, similar to the product expansion of the Carlitz period. We further link to results of Gekeler and Maurischat on the $\infty $ -adic field generated by the period lattice.

Colmez [Périodes des variétés abéliennes a multiplication complexe, Ann. of Math. (2)138(3) (1993), 625–683; available at http://www.math.jussieu.fr/∼colmez] conjectured a product formula for periods of abelian varieties over number fields with complex multiplication and proved it in some cases. His conjecture is equivalent to a formula for the Faltings height of CM abelian varieties in terms of the logarithmic derivatives at $s=0$ of certain Artin $L$-functions. In a series of articles we investigate the analog of Colmez’s theory in the arithmetic of function fields. There abelian varieties are replaced by Drinfeld modules and their higher-dimensional generalizations, so-called $A$-motives. In the present article we prove the product formula for the Carlitz module and we compute the valuations of the periods of a CM $A$-motive at all finite places in terms of Artin $L$-series. The latter is achieved by investigating the local shtukas associated with the $A$-motive.

Let $\psi$ be a generic Drinfeld module of rank $r\,\ge \,2$ . We study the first elementary divisor ${{d}_{1,\,\wp }}\,\left( \psi\right)$ of the reduction of $\psi$ modulo a prime $\wp $ , as $\wp $ varies. In particular, we prove the existence of the density of the primes $\wp $ for which ${{d}_{1,\,\wp }}\,\left( \psi\right)$ is fixed. For $r\,=\,2$ , we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp $ and prove that, on average, it has a large norm. Our work is motivated by J.-P. Serre's study of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M. R. Murty.

We formulate and prove a Sato–Tate equidistribution law for Drinfeld modules.

We study the transcendence degree (of the values at algebraic points) of the coordinate functions of a given one parameter subgroup on a simple g dimensional T-module. In such a situation, we obtain a lower bound for this transcendence degree depending on g and on the growth order of the parameter subgroup. As a particular case, one gets the finite characteristic analogue of a result of Siegel on the algebraic independence for the values of the classical Bessel function.

Résumé.  Nous obtenons une minoration du degré de transcendance d'un ensemble de valeurs en des points algébriques des fonctions coordonnées d'un sous-groupe à un paramètre d'un T-module simple, en fonction de l'ordre de croissance de ces dernières et de la dimension du T-module. Un cas particulier de ce résultat permet d'obtenir en caractéristique finie un analogue d'un résultat de Siegel sur l'indépendance algébrique des valeurs de la fonction de Bessel usuelle.