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.

Let $k = {\Bbb F}_q(t)$ be the rational function field with finite constant field and characteristic $p \geq 3$, and let K/k be a finite separable extension. For a fixed place v of k and an elliptic curve E/K which has ordinary reduction at all places of K extending v, we consider a canonical height pairing $\langle \phantom {x},\phantom {x}\! \rangle _v \colon E(K^{\rm {sep}}) \times E(K^{\rm {sep}}) \to {\Bbb C}_{v}^{\times }$ which is symmetric, bilinear and Galois equivariant. The pairing $\langle \phantom {x},\phantom {x}\! \rangle _\infty$ for the “infinite” place of k is a natural extension of the classical Néron–Tate height. For v finite, the pairing $\langle \phantom {x},\phantom {x}\! \rangle _v$ plays the role of global analytic p-adic heights. We further determine some hypotheses for the nondegeneracy of these pairings.