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 $F^{\infty }=F\left(E\left[p^{\infty }\right]\right)$, where $E\left[p^{\infty }\right]$ is the group of $p$-power torsion points of $E$. Let $\tau$ be an irreducible self-dual complex representation of $Gal\left(F^{\infty }∕F\right)$. With certain assumptions on $E$ and $p$, we give explicit formulas for the root number $W\left(E,\tau \right)$. 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 $GL\left(2,\mathcal{O}\right)$, where $\mathcal{O}$ is the ring of integers of a finite extension of ${\mathbb{Q}}_p$, for $p$ an odd prime. Among all such representations, those that factor through $PGL\left(2,\mathcal{O}\right)$ have been analyzed in detail in existing literature. We give a complete description of those irreducible self-dual complex representations of $GL\left(2,\mathcal{O}\right)$ that do not factor through $PGL\left(2,\mathcal{O}\right)$.