Rank-Two Drinfeld Modules over Elliptic Coordinate Rings

Abstract

Let $\mathbf{A}$ be the affine coordinate ring of $E\setminus \{\infty \}$, where $E$ is an elliptic curve over ${\mathbb{F}}_q$ and $\infty$ is rational. We solve the rank-two case of the open problem of explicitly classifying sign-normalized Drinfeld $\mathbf{A}$-modules in generic characteristic with a fixed determinant motive; the classification for rank greater than two remains open. Outside an explicit exceptional locus, the rank-two Drinfeld modules are parametrized by the open modular curve $Y^{q+1}=\pi (X)$ with $Y\ne 0$. The polynomial $\pi$ is squarefree, and the smooth projective completion of this curve has genus $q^2$. The roots of $\pi$ split into two types. Above roots of the first type, all coefficient functions extend regularly, and their extensions yield exactly the exceptional modules. Roots of the second type correspond to cusps of the modular curve; above these roots, the coefficient family has poles.

arXiv comment32 pages