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Representation theory and the cuspidal group of \(X(p)\). (English) Zbl 0629.14021

The author uses the characteristic p representation theory of \(\mathrm{GL}_ 2(\mathbb Z/p\mathbb Z)\) to study the structure of the \(p\)-part of the cuspidal divisor class group \(C\) of the principal modular curve \(X(p)\) as a module over the Galois group of \(X(p)\). In the process he recovers several results of D. S. Kubert and S. Lang [“Modular units.” New York etc.: Springer (1981; Zbl 0492.12002)]. In particular, he sees that the existence of the special group is required by the non-semisimplicity of the principal series representations of \(\mathrm{GL}_ 2(\mathbb Z/p\mathbb Z)\) in characteristic \(p\). In the same vein, the fact that certain modular representations are not semisimple is used to show that, when \(p\geq 5\), the quotient \(C/pC\) is of dimension \(\geq (p-5)(p-1)/4\) with equality if and only if \(p\) is a regular prime.
Reviewer: S. Kamienny

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14G25 Global ground fields in algebraic geometry

Citations:

Zbl 0492.12002
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References:

[1] E. Hecke, Grundlagen einer Theorie der Integralgruppen und der Integralperioden bei den Normalteilern der Modulgruppe , Math. Ann. 116 (1939), 469-510, ( = Math. Werke 38, 731-772). · Zbl 0022.11002
[2] D. S. Kubert and S. Lang, Modular Units , Grundlehren der Mathematischen Wissenschaften, vol. 244, Springer-Verlag, New York, 1981. · Zbl 0492.12002
[3] Ju. I. Manin, Parabolic points and zeta functions of modular curves , Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19-66, (AMS Translations). · Zbl 0243.14008
[4] J.-P. Serre, Linear Representations of Finite Groups , Graduate Texts in Mathematics, vol. 42, Springer-Verlag, New York, 1977. · Zbl 0355.20006
[5] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions , Princeton University Press, 1971. · Zbl 0221.10029
[6] R. Steinberg, Lectures on Chevalley Groups , Yale University Press, 1968. · Zbl 0307.22001
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