Triply imprimitive representations of GL(2)

Abstract

We give a criterion for an irreducible, admissible, supercuspidal representation π \pi of G L ( 2 , K ) \mathrm {GL}(2,K) , where K K is a p p -adic field, to become a principal series representation under every quadratic base change. We determine all such π \pi that have trivial central character and conductor 2 2 , and explain their relevance for the theory of elliptic curves.