FRACTIONAL POWER SERIES AND

PAIRINGS ON DRINFELD MODULES

Bjorn Poonen

Abstract. Let C be an algebraically closed field containing the finite field F_{q} and complete with respect to an absolute value | |. We prove that under suitable constraints on the coefficients, the series f(z) = ∑_{n∈Z} a_{n}z^{q^{n}} converges to a surjective, open, continuous F_{q}-linear homomorphism C → C whose kernel is locally compact. We characterize the locally compact sub-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^{∗}(z) = ∑_{n∈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 × ker f^{∗} → F_{q}

which exhibits ker f^{∗} 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.

Typeset by AMS-TEX