Semistable deformation rings in even Hodge-Tate weights
Let \(\bar{\rho} : \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(\bar{\mathbb{F}}_p)\) be a continuous representation. \textit{C. Breuil} and \textit{A. Mézard} [Duke Math. J. 115, No. 2, 205--310 (2002; Zbl 1042.11030)] computed the irreducible components of the semistable deformation ring for \(\bar{\rho}\) with Hodge-Tate weights \((0,r)\), where \(0 < r < p-1\) is odd. They also computed the mod \(p\) reductions of semistable representations with these Hodge-Tate weights in terms of their parameters: a Frobenius eigenvalue \(\lambda\) and the \(\mathfrak{L}\)-invariant. The present paper carries out this calculation in the case that \(0 < r< p-1\) is even. The method is as in [loc. cit.]: to write down `strongly divisible modules' corresponding to Galois-stable lattices in semistable Galois representations with given parameters. This is a very involved calculation, even more so when \(r\) is even. One novelty is that the strongly divisible modules in the even case require coefficients that are not simply rational functions of \(\lambda\) and \(\mathfrak{L}\), but instead are limits of such functions satisfying certain quadratic equations.
scientific article; zbMATH DE number 7050832
scientific article