Twists of Galois Representations and Projective Automorphisms, Original Research Article

We show that two surjectiveλ-adic Galois representations which areλ-adically close near the supersingular primes are equivalent up to a twist and a standard automorphism ofGLn. In particular, two elliptic curves 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 ofPGLnover a complete local ring.