Computation of Framed Deformation Functors

In this work we compute the universal framed deformation functor for a reducible Galois representation $\rho$ given by direct sum of 2-dimensional representations ${\rho }_i$ coming from p-divisible groups. We impose the local conditions of flatness in the residual characteristic prime $p$ and semistable action in a single auxiliary prime $\mathrm{\ell }$. The main application is the case of ${\rho }_i$ being the representation attached to the $p$-torsion points of an elliptic curve $E_i$ over $\mathbb{Q}$ with good reduction but in $\mathrm{\ell }$ and semistable reduction in $\mathrm{\ell }$. This article was inspired by a series of works of Schoof about abelian varieties with bad reduction in one prime only. Our local deformation condition correspond to reduction properties of such varieties.