Representation theory and the cuspidal group of \(X(p)\) (Q1093695)

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 \textit{D. S. Kubert} and \textit{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.