Elementary computation of ramified components of the Jacobi sum Hecke characters (Q983295)

In [J. Reine Angew. Math. 385, 41--101 (1988; Zbl 0654.12003)], \textit{R. Coleman} and \textit{W. McCallum} have calculated the ramified components of the Jacobi sum Hecke characters. For this purpose they had to compute explicitely the stable reduction of the Fermat curve. The author gives a short and simple proof of the result. He computes the Galois action on the étale cohomology group \(H^1_{c}(U_{\bar{K}},\mathcal{K}_{X})\), where \(U\) is the projective line minus \(0,1,\infty\) and \(\mathcal{K}_{X}\) a smooth \(\bar{\mathbf Q}_{\ell}\)-sheaf associated to the quotient of the Fermat curve, like in [\textit{ibid.}], but does not need to use a stable model of the Fermat curve, hence any rigid geometry.