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Journal of the American Mathematical Society

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

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Fractional Power Series and Pairings on Drinfeld Modules
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by Bjorn Poonen PDF
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.
References
  • Yvette Amice, Les nombres $p$-adiques, Collection SUP: “Le Mathématicien”, vol. 14, Presses Universitaires de France, Paris, 1975 (French). Préface de Ch. Pisot. MR 0447195
  • David L. Armacost, The structure of locally compact abelian groups, Monographs and Textbooks in Pure and Applied Mathematics, vol. 68, Marcel Dekker, Inc., New York, 1981. MR 637201
  • C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
  • Pierre Deligne and Dale Husemoller, Survey of Drinfel′d modules, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 25–91. MR 902591, DOI 10.1090/conm/067/902591
  • Martin Eichler, Introduction to the theory of algebraic numbers and functions, Pure and Applied Mathematics, Vol. 23, Academic Press, New York-London, 1966. Translated from the German by George Striker. MR 0209258
  • N. Elkies: Linearized algebra and finite groups of Lie type, preprint, 1994.
  • Joe Flood, Pontryagin duality for topological modules, Proc. Amer. Math. Soc. 75 (1979), no. 2, 329–333. MR 532161, DOI 10.1090/S0002-9939-1979-0532161-7
  • Catherine Goldstein (ed.), Séminaire de Théorie des Nombres, Paris 1988–1989, Progress in Mathematics, vol. 91, Birkhäuser Boston, Inc., Boston, MA, 1990. Papers from the seminar held in Paris, 1988–1989. MR 1104695, DOI 10.1007/978-1-4612-3460-9
  • Irving Glicksberg, Uniform boundedness for groups, Canadian J. Math. 14 (1962), 269–276. MR 155923, DOI 10.4153/CJM-1962-017-3
  • D. Goss: The adjoint of the Carlitz module and Fermat’s Last Theorem, preprint, 1994.
  • Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
  • Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR 719020
  • Kneser, M.: Algebraische Zahlentheorie, Vorlesungsausarbeitung Georg-August-Universität Göttingen, 1966.
  • Neal Koblitz, $p$-adic numbers, $p$-adic analysis, and zeta-functions, 2nd ed., Graduate Texts in Mathematics, vol. 58, Springer-Verlag, New York, 1984. MR 754003, DOI 10.1007/978-1-4612-1112-9
  • Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
  • J. Neukirch: Algebraische Zahlentheorie, Springer-Verlag, 1992.
  • Ø. Ore: On a Special Class of Polynomials, Trans. Amer. Math. Soc. 35 (1933), 559–584.
  • Yuichiro Taguchi, Semi-simplicity of the Galois representations attached to Drinfel′d modules over fields of “infinite characteristics”, J. Number Theory 44 (1993), no. 3, 292–314. MR 1233291, DOI 10.1006/jnth.1993.1055
  • André Weil, Basic number theory, 3rd ed., Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer-Verlag, New York-Berlin, 1974. MR 0427267, DOI 10.1007/978-3-642-61945-8
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Additional Information
  • 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.
  • © Copyright 1996 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 9 (1996), 783-812
  • MSC (1991): Primary 13J05; Secondary 11G09
  • DOI: https://doi.org/10.1090/S0894-0347-96-00203-2
  • MathSciNet review: 1333295