Computation of Framed Deformation Functors

TL;DR

This paper computes the universal framed deformation functor for certain reducible Galois representations arising from p-divisible groups, with specific local conditions related to elliptic curves and reduction properties.

Contribution

It introduces a method to compute deformation functors for reducible Galois representations with particular local conditions, extending previous work on abelian varieties with bad reduction.

Findings

Explicit computation of the universal framed deformation functor.

Application to Galois representations from elliptic curves with specific reduction properties.

Connection to Schoof's work on abelian varieties with bad reduction.

Abstract

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 $\ell$. 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 $\ell$ and semistable reduction in $\ell$. 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.