Abstract
Let H be the upper half-plane and Γ a subgroup of SL2(Z) containing some principal congruence subgroup Γ(N). Then the Riemann surface Γ\H, compactified by the cusps of Γ, is the set of C-valued points of a complete algebraic curve MC (Γ) defined over Q(e2πi/N) (and often over a smaller field). One knows that MC(Γ) has good reduction at primes not dividing N, but it is only recently that anything general has been known about the reduction at other primes. Suppose that N = q · Q, where q is a prime not dividing Q. Recall that
and let
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
W. CASSELMAN: On abelian varieties with many endomorphisms and a conjecture of Shimura’s, Inventiones Math. 12 (1971), 225–236.
([AR]): Assorted results on representations of GL2(k), in this volume.
M. EICHLER: Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion, Arch. Math. 5 (1954), 355–366.
I.M. GELFAND, M.I. GRAEV, I.I. PYATETSKII-SHAPIRO: Representation Theory and Automorphic Functions (translated from Russian), W.B. Saunders, Philadelphia, 1969.
E. HECKE: Über das Verhalten der Integrale 1. Gattung bei Abbildungen, insbesandere in der Theorie der elliptischen Modulfunktionen, Werke No. 29.
E. HECKE: Gundlagen einen Theorie der Integralgruppen und der Integralperioden bei den Normalteilern der Modulgruppe, Werke No. 38, Vandenhoeck and Ruprecht, Göttingen, 1970.
H. JACQUET and R.P. LANGLANDS: Automorphic forms for GL2, Springer Lecture Notes 114, 1970.
M. KNESER: Strong approximation, in Algebraic Groups and Discontinuous Subgroups, American Math. Society, Providence, 1966.
R.P. LANGLANDS: On the functional equation of the Artin L-functions, Yale University Lecture Notes.
R.P. LANGLANDS: Problem in the theory of automorphic forms, Yale University Lecture Notes.
A. OGG: Elliptic Curves and Wild Ramification, Amer. Jour. Math. 89 (1967), 1–21.
P. SALLY and J. SHALIKA: The Plancherel formula for SL(2) over a local field, P.N.A.S. 63 (1969), 661–667.
J-P. SERRE: Cohomologie Galoisienne, Springer Lecture Notes 5, 1964.
J-P. SERRE: Abelian ℓ-adic Representations and Elliptic Curves, Benjamin, 1968.
J-P. SERRE and J. TATE: Good reduction of abelian varieties, Ann. of Math. 88 (1968), 492–517.
G. SHIMURA: Correspondences modulaires et les fonctions ζ de courbes algébriques, J. Math. Soc. Japan 10 (1958), 1–28.
G. SHIMURA: Arithmetic Theory of Automorphic Functions, Princeton, 1971).
G. SHIMURA: On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields, Nagoya Math. Jour. 43 (1971), 199–208.
J. VÉLU: Courbes elliptiques sur Q ayant bonne réduction en dehors de {11}, C.R. Acad. Sc. Paris 273 (1971), 73–75.
A. WEIL: Sur certains groupes d’opérateurs unitaires, Acta. Math. 111 (1964), 143–211.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1973 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Casselman, W. (1973). On Representations of Gl2 and The Arithmetic of Modular Curves. In: Deligne, P., Kuijk, W. (eds) Modular Functions of One Variable II. Lecture Notes in Mathematics, vol 349. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-37855-6_3
Download citation
DOI: https://doi.org/10.1007/978-3-540-37855-6_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06558-6
Online ISBN: 978-3-540-37855-6
eBook Packages: Springer Book Archive