Twisted root numbers of elliptic curves semistable at primes above 2 and 3

Abstract

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 nonabelian. 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 ℚp, for p an odd prime. Among all such representations, those that factor through PGL(2,O) have been analyzed in detail in existing literature. We give a complete description of those irreducible self-dual complex representations of GL(2,O) that do not factor through PGL(2,O).