# Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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## Fractional Power Series and Pairings on Drinfeld ModulesHTML articles powered by AMS MathViewer

by Bjorn Poonen
J. Amer. Math. Soc. 9 (1996), 783-812 Request permission

## 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.
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• Bjorn Poonen
• Affiliation: Mathematical Sciences Research Institute, Berkeley, California 94720-5070
• Address at time of publication: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000
• MR Author ID: 250625
• ORCID: 0000-0002-8593-2792
• Email: poonen@msri.org, poonen@math.princeton.edu
• Received by editor(s): December 9, 1994
• Received by editor(s) in revised form: May 22, 1995
• Additional Notes: This research was supported by a Sloan Doctoral Dissertation Fellowship.