### 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 <shekhar@math.tifr.res.in>