Nonvanishing of all exponential coefficients

Determine whether every exponential coefficient e_n(\varphi) of the exponential series of an arbitrary rank-one Drinfeld A-module over the curve is nonzero.

Background

For a rank-one Drinfeld A-module \varphi, the exponential function has an expansion $\exp_\varphi(z)=\sum_{n\ge 0}e_n(\varphi)z^{q^n}$. The paper establishes restrictions on possible vanishing: Thakur's results imply that a zero can occur only for indices $n\in\{2,\ldots,g-1\}$, where $g$ is the genus, and that no such vanishing occurs when the function field has a degree-one place. The coefficients are also known to be nonzero in the Carlitz and elliptic examples.

The unresolved issue is whether vanishing can occur at all in the remaining higher-genus cases. Resolving it would establish the expected nonvanishing of every exponential coefficient and would justify the reciprocal notation based on evaluations of the Frobenius products at the characteristic orbit in full generality.