$m$-bigness in compatible systems

Details $m$-bigness in compatible systems $m$-bigness in compatible systems

2010-07-02

arxiv.org/abs/1007.0358v2

C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1049-1054

Mathematics

Number Theory

Scientific paper

10.1016/j.crma.2010.09.001

Taylor-Wiles type lifting theorems allow one to deduce that for $\rho$ a "sufficiently nice" $l$-adic representation of the absolute Galois group of a number field whose semi-simplified reduction modulo $l$, denoted $\overline{\rho}$, comes from an automorphic representation then so does $\rho$. The recent lifting theorems of Barnet-Lamb-Gee-Geraghty-Taylor impose a technical condition, called \emph{$m$-big}, upon the residual representation $\overline{\rho}$. Snowden-Wiles proved that for a sufficiently irreducible compatible system of Galois representations, the residual images are \emph{big} at a set of places of Dirichlet density $1$. We demonstrate the analogous result in the \emph{$m$-big} setting using a mild generalization of their argument.

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