Gross, Benedict H. Representation theory and the cuspidal group of \(X(p)\). (English) Zbl 0629.14021 Duke Math. J. 54, 67-75 (1987). 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 Cited in 3 Documents MSC: 11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields 14G25 Global ground fields in algebraic geometry Keywords:characteristic p representation; principal modular curve Citations:Zbl 0492.12002 × Cite Format Result Cite Review PDF Full Text: DOI 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 · doi:10.1007/BF01597370 [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 · doi:10.1070/IM1972v006n01ABEH001867 [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.