Twists of Galois representations and projective automorphisms

We show that two surjective lambda-adic Galois representations which are la mbda-adically close near the supersingular primes are equivalent up to a tw ist and a standard automorphism of GL(n). In particular, two elliptic curve s over a number field which are locally twist of each other in fact differ by a global twist. The proof depends on determining the automorphisms of PG L(n) over a complete local ring. (C) 1999 Academic Press.