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Mathematics of Computation

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Rank-one Drinfel'd modules on elliptic curves

Authors: D. S. Dummit and David Hayes
Journal: Math. Comp. 62 (1994), 875-883
MSC: Primary 11G09; Secondary 11G15
MathSciNet review: 1218342
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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.

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Article copyright: © Copyright 1994 American Mathematical Society

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