Are supercuspidal reps of GL(2) uniquely determined by the rootnumber

Question

Let $\pi, \pi'$ be a unitary, irreducible, supercuspidal representations of $GL_2(F)$. Does an equality of roots numbers $\epsilon(\pi, \psi, s) = \epsilon(\pi', \psi, s)$ for all $s \in \mathbb{C}$ imply an isomorphism $\pi \cong \pi'$?

Answer 1

No -- there are far too many supercuspidals $\pi$ for it to be possible to distinguish them by a single $\varepsilon$-factor -- but this is true if you also consider root numbers of twists; this is called the "local converse theorem" and is in the Jacquet-Langlands book (SLN 114). See also Andrew Snowden's thesis. There is a generalization to $GL(n)$ if you allow twists by $GL(n-2)$ -- this is a result of Henniart.