Vol. 4, No. 3, 2010

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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

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