Vol. 4, No. 3, 2010

Download this article
Download this article For screen
For printing
Recent Issues

Volume 11
Issue 10, 2213–2445
Issue 9, 1967–2212
Issue 8, 1739–1965
Issue 7, 1489–1738
Issue 6, 1243–1488
Issue 5, 1009–1241
Issue 4, 767–1007
Issue 3, 505–765
Issue 2, 253–503
Issue 1, 1–252

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors' Addresses
Editors' Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Author Index
To Appear
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Twisted root numbers of elliptic curves semistable at primes above 2 and 3

Ryota Matsuura

Vol. 4 (2010), No. 3, 255–295

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(FF). 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 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).

elliptic curves, root number, Mordell–Weil rank
Mathematical Subject Classification 2000
Primary: 11G05
Secondary: 11F80, 11G40
Received: 15 January 2009
Revised: 28 November 2009
Accepted: 29 November 2009
Published: 5 February 2010
Ryota Matsuura
School of Education
Boston University
Two Silber Way
Boston, MA 02215
United States