Fractional Power Series and Pairings on Drinfeld Modules

Abstract:

Let $C$ be an algebraically closed field containing $\mathbb {F}_q$ which is complete with respect to an absolute value $|\;|$. We prove that under suitable constraints on the coefficients, the series $f(z) = \sum _{n \in \mathbb {Z}} a_n z^{q^n}$ converges to a surjective, open, continuous $\mathbb {F}_q$-linear homomorphism $C \rightarrow C$ whose kernel is locally compact. We characterize the locally compact sub-$\mathbb {F}_q$-vector spaces $G$ of $C$ which occur as kernels of such series, and describe the extent to which $G$ determines the series. We develop a theory of Newton polygons for these series which lets us compute the Haar measure of the set of zeros of $f$ of a given valuation, given the valuations of the coefficients. The “adjoint” series $f^\ast (z) = \sum _{n \in \mathbb {Z}} a_n^{1/q^n} z^{1/q^n}$ converges everywhere if and only if $f$ does, and in this case there is a natural bilinear pairing \[ \ker f \times \ker f^\ast \rightarrow \mathbb {F}_q \] which exhibits $\ker f^\ast$ as the Pontryagin dual of $\ker f$. Many of these results extend to non-linear fractional power series. We apply these results to construct a Drinfeld module analogue of the Weil pairing, and to describe the topological module structure of the kernel of the adjoint exponential of a Drinfeld module.