Limits of residually irreducible p-adic Galois representations, by Chandrashekhar Khare

A corrected version of this paper appears as preprint 0289. We study sequences of n-dimensional p-adic representations of Galois groups of number fields that ``converge''. We prove that if these representations are assumed to be motivic of fixed weight then the degree of the fields of rationality of such (non-constant) converging sequences tends to infinity. Using Ramakrishna's lifting theorems this gives some useful information about rationality of lifts of 2-dimensional odd representations of the Galois group of Q. We also determine the p-adic closure of the set of lifts produced by Ramakrishna, and raise some questions about the relationship between algebraicity of p-adic representations and their ramification properties.

Chandrashekhar Khare <>