Root numbers of elliptic curves, semistable at primes above 2 and 3

Let E be an elliptic curve over a number field F, and fix a rational prime p. Put F^∞=F(E[p^∞]) where E[p^∞] is the group of p-power torsion points of E. Let τ be an irreducible self-dual complex representation of Gal(F^∞/F). With certain assumptions on E and p, we give explicit formulas for the root number W(E,τ). We use these root numbers to study the growth of the rank of E in its own division tower and also to count the trivial zeros of the L-function of E. Moreover, our assumptions ensure that the p-division tower of E is non-abelian. In the process of computing the root number, we also study the irreducible self-dual complex representations of GL(2,O) where O is the ring of integers of a finite extension of Q_p, for p an odd prime. Among all such representations, those that factor through PGL(2,O) have been analyzed in detail in existing literature. In our work, we give a complete description of those irreducible self-dual complex representations of GL(2,O) that do not factor through PGL(2,O).