Tame inertia weights of certain crystalline representations (Q614476)

Let \(V\) be a \(p\)-adic representation of \(\mathrm{Gal}(K^{\mathrm{alg}}/K)\) where \(K\) is a \(p\)-adic field, and let \(\overline{V}\) be the ``reduction modulo \(p\)'' of \(V\). In their article ``Hodge and Newton and tame inertia polygons of semi-stable representations'' [Math. Ann. 343, No. 4, 773--789 (2009; Zbl 1248.11092)], the authors explain how to attach a polygon to \(\overline{V}\), constructed from the weights of the tame inertia subgroup of \(\mathrm{Gal}(K^{\mathrm{alg}}/K)\) acting on \(\overline{V}\). They also proved that under certain conditions, this polygon is above the Hodge polygon which can be constructed from the Hodge-Tate weights of \(V\), with the same endpoints. The question then arises as to which polygons, lying above the Hodge polygon and having the same endpoints, can actually occur as the tame inertia polygon. In the article under review, the authors determine these polygons for a family of crystalline representations of dimension \(2\) of \(\mathrm{Gal}(K^{\mathrm{alg}}/K)\), where \(K\) is allowed some ramification. In particular, they observe that the tame inertia polygon and the Hodge polygon need not be the same.