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

Matsuura, Ryota

doi:10.2140/ant.2010.4.255

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

Ryota Matsuura

Vol. 4 (2010), No. 3, 255–295

DOI: 10.2140/ant.2010.4.255

Abstract

Let $E$ be an elliptic curve over a number field $F$ , and fix a rational prime $p$ . Put $F^{\infty} = F (E [p^{\infty}])$ , where $E [p^{\infty}]$ is the group of $p$ -power torsion points of $E$ . Let $τ$ be an irreducible self-dual complex representation of $Gal (F^{\infty} ∕ F)$ . 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 $ℚ_{p}$ , for $p$ an odd prime. Among all such representations, those that factor through $PGL (2, O)$ have been analyzed in detail in existing literature. We give a complete description of those irreducible self-dual complex representations of $GL (2, O)$ that do not factor through $PGL (2, O)$ .

Keywords

elliptic curves, root number, Mordell–Weil rank

Mathematical Subject Classification 2000

Primary: 11G05

Secondary: 11F80, 11G40

Milestones

Received: 15 January 2009

Revised: 28 November 2009

Accepted: 29 November 2009

Published: 5 February 2010

Authors

Ryota Matsuura
School of Education Boston University Two Silber Way Boston, MA 02215 United States

Vol. 4, No. 3, 2010