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Frobenius distributions of Drinfeld modules over finite fields


Author: Ernst-Ulrich Gekeler
Journal: Trans. Amer. Math. Soc. 360 (2008), 1695-1721
MSC (2000): Primary 11G09
DOI: https://doi.org/10.1090/S0002-9947-07-04558-8
Published electronically: November 26, 2007
MathSciNet review: 2366959
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Abstract: We express the weighted class number of Drinfeld $ A$-modules of rank two with given characteristic polynomial over the finite field $ {\mathbb{F}} _{\mathfrak{p}}=A/{\mathfrak{p}}$ $ ({\mathfrak{p}} \in\operatorname{Spec}A$, where $ A=\mathbb{F} _q[T])$ as an infinite product of local terms. Some auxiliary results of independent interest about characteristic polynomials of Drinfeld modules are given.


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  • [1] David, C.: Frobenius distributions of Drinfeld modules of any rank, J. Numb. Th. 90 (2001),329-340. MR 1858082 (2002k:11084)
  • [2] Deligne, P., Husemöller, D.: Survey of Drinfeld modules, Contemp. Math. 67 (1987), 25-91. MR 902591 (89f:11081)
  • [3] Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hamburg 14 (1941), 197-272. MR 0005125 (3:104f)
  • [4] Drinfeld, V.G.: Elliptic modules (Russian), Math. Sbornik 94 (1974), 594-627, English translation: Math. USSR-Sbornik 23 (1976), 561-592. MR 0384707 (52:5580)
  • [5] Drinfeld, V.G.: Elliptic modules II, Math. USSR-Sbornik 31 (1977), 159-170.
  • [6] Gekeler, E.-U.: Zur Arithmetik von Drinfeld-Moduln, Math. Ann. 262 (1983), 167-182. MR 690193 (84j:12010)
  • [7] Gekeler, E.-U.: Über Drinfeld'sche Modulkurven vom Hecke-Typ, Comp. Math. 57 (1986), 219-236. MR 827352 (87d:11041)
  • [8] Gekeler, E.-U.: On the coefficients of Drinfeld modular forms, Invent. Math. 93 (1988), 667-700. MR 952287 (89g:11043)
  • [9] Gekeler, E.-U.: On finite Drinfeld modules, J. Algebra 141 (1991), 187-203. MR 1118323 (92e:11064)
  • [10] Gekeler, E.-U.: Highly ramified pencils of elliptic curves in characteristic two, Duke Math. J. 89 (1997), 95-107. MR 1458973 (99d:11063)
  • [11] Gekeler, E.-U.: Frobenius distributions of elliptic curves over finite prime fields, Int. Math. Res. Notes 37 (2003), 1999-2018. MR 1995144 (2004d:11048)
  • [12] Goss, D.: Basic structures of function field arithmetic, Springer-Verlag 1996. MR 1423131 (97i:11062)
  • [13] Hsia, L.-Ch., Yu, J.: On characteristic polynomials of geometric Frobenius associated to Drinfeld modules, Comp. Math. 122 (2000), 261-280. MR 1781330 (2001h:11119)
  • [14] Jung, F.: Charakteristische Polynome von Drinfeld-Moduln, Diplomarbeit Saarbrücken 2000.
  • [15] Neukirch, J.: Class field theory, Springer-Verlag, 1986. MR 819231 (87i:11005)
  • [16] Rosen, M.: Number theory in function fields, Springer-Verlag, New York, 2002. MR 1876657 (2003d:11171)
  • [17] Schweizer, A.: On Drinfeld modular curves with many rational points over finite fields, Finite Fields Appl. 8 (2002), 434-443. MR 1933615 (2004c:11096)
  • [18] Shimura, G.: Arithmetic theory of automorphic functions, Princeton University Press, 1971.
  • [19] Yu, J.-K.: Isogenies of Drinfeld modules over finite fields, J. Number Th. 54 (1995), 161-171. MR 1352643 (96i:11060)
  • [20] Yu, J.-K.: A Sato-Tate law for Drinfeld modules, Comp. Math. 138 (2003), 189-197. MR 2018826 (2005a:11084)
  • [21] Drinfeld modules, modular schemes and applications, Proc. Alden-Biesen 1996, E.-U. Gekeler et al. (eds.), World Scientific 1997. MR 1630594 (99b:11002)

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Additional Information

Ernst-Ulrich Gekeler
Affiliation: FR 6.1 Mathematik, Universität des Saarlandes, Postfach 15 11 50, D-66041 Saar- brücken, Germany
Email: gekeler@math.uni-sb.de

DOI: https://doi.org/10.1090/S0002-9947-07-04558-8
Received by editor(s): March 16, 2005
Published electronically: November 26, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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