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ISSN 1088-6850(e) ISSN 0002-9947(p)

Frobenius distributions of Drinfeld modules over finite fields

Author(s): Ernst-Ulrich Gekeler
Journal: Trans. Amer. Math. Soc. 360 (2008), 1695-1721.
MSC (2000): Primary 11G09
Posted: November 26, 2007
MathSciNet review: 2366959
Retrieve article in: PDF

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Abstract: We express the weighted class number of Drinfeld -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.

References:

[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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