Rank-one Drinfel’d modules on elliptic curves

Abstract:

The sgn-normalized rank-one Drinfeld modules $\phi$ associated with all elliptic curves E over ${\mathbb {F}_q}$ for $4 \leq q \leq 13$ are computed in explicit form. (Such $\phi$ for $q < 4$ were computed previously.) These computations verify a conjecture of Dorman on the norm of $j(\phi ) = {a^{q + 1}}$ and also suggest some interesting new properties of $\phi$. We prove Dorman’s conjecture in the ramified case. We also prove the formula $\deg N(a) = q({h_k} - 1 + q)$, where $N(a)$ is the norm of a and ${h_k}$ is the class number of $k = {\mathbb {F}_q}(E)$. We describe a remarkable conjectural property of the trace of a in even characteristic that holds in all the examples.