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Gekeler, Ernst-Ulrich

Über Drinfeld’sche Modulkurven vom Hecke-Typ. (On Drinfel’d modular curves of Hecke type). (German) Zbl 0599.14032

Compos. Math. 57, 219-236 (1986).
The purpose of this paper is to investigate the Drinfeld modular curve \(X_ 0(n)\) by reduction mod(n), where (n) denotes a finite prime place of rational function field over a finite field. Among others, the author evaluates the order of the cusp divisor class of \(X_ 0(n)\) (in § 4) using the idea of Ogg and the results of Raynaud [A. P. Ogg, Analytic number theory, Proc. Symp. Pure Math. 24, St. Louis Univ., Missouri 1982, 211-231 (1973; Zbl 0273.14008), M. Raynaud, Publ. Math., Inst. Hautes Étud. Sci. 38, 27-76 (1970; Zbl 0207.51602)].
Reviewer: K.Katayama

Cited in 3 Reviews
Cited in 28 Documents

MSC:

14H45 Special algebraic curves and curves of low genus
11G20 Curves over finite and local fields
11G18 Arithmetic aspects of modular and Shimura varieties
11R58 Arithmetic theory of algebraic function fields
14G15 Finite ground fields in algebraic geometry

Keywords:

finite ground field; Drinfeld modular curve; order of the cusp divisor class; Atkin-Lehner involution

Citations:

Zbl 0273.14008; Zbl 0207.51602
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\textit{E.-U. Gekeler}, Compos. Math. 57, 219--236 (1986; Zbl 0599.14032)
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References:

[1] A. Brumer : Courbes modulaires , Grenoble 1975.
[2] P. Deligne and D. Mumford : The irreducibility of the space of curves of given genus , Publ. IHES no. 36 (1969) 75-110. · Zbl 0181.48803
[3] P. Deligne and M. Rapoport : Les schémas de modules de courbes elliptiques . In: Modular Forms of One Variable II , Lecture Notes in Mathematics, Vol. 349, Berlin- Heidelberg-New York: Springer (1973). · Zbl 0281.14010
[4] V.G. Drinfeld : Elliptic modules , Math. USSR-Sbornik 23 (1976) 561-592. · Zbl 0386.20022
[5] E. Gekeler : Drinfeld-Moduln und modulare Formen über rationalen Funktionen-körpern . Bonner Math. Schriften 119 (1980). · Zbl 0446.14018
[6] E. Gekeler : Zur Arithmetik von Drinfeld-Moduln , Math. Annalen 256 (1982) 549-560. · Zbl 0536.14028
[7] E. Gekeler : A Product Expansion for the Discriminant Function of Drinfeld Modules Jour . Number Theory 21 (1985) 135-140. · Zbl 0572.10021
[8] E. Gekeler : Modulare Einheiten für Funktionenkörper , Crelle’s Journal 348 (1984) 94-115. · Zbl 0523.14021
[9] E. Gekeler : Automorphe Formen über Fq(T) mit kleinem Führer (in Vorbereitung) .
[10] D. Goss : \pi -adic Eisenstein series for function fields , Comp. Math. 41 (1980) 3-38. · Zbl 0388.10020
[11] D. Goss : The algebraist’s upper half-plane , Bull. Amer. Math. Soc. 2 (1980) 391-415. · Zbl 0433.14017
[12] D. Goss : Modular forms for Fr[T] , Crelle’s Journal 231 (1980) 16-39. · Zbl 0422.10021
[13] D. Hayes : Explicit class field theory for rational function fields , Trans. Amer. Math. Soc. 189 (1974) 77-91. · Zbl 0292.12018
[14] J. Lipman : Rational singularities with applications to algebraic surfaces and unique factorization , Publ. IHES no. 36 (1969) 195-280. · Zbl 0181.48903
[15] B. Mazur : Modular curves and the Eisenstein ideal , Publ. IHES no. 47 (1977) p. 33-186. · Zbl 0394.14008
[16] A. Ogg : Rational Points on Certain Elliptic Modular Curves , AMS Conference St. Louis (1972) 221-231. · Zbl 0273.14008
[17] M. Raynaud : Spécialisation du Foncteur de Picard , Publ. IHES no. 38 (1970) 27-76. · Zbl 0207.51602
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