## Algebra & Number Theory

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

Ryota Matsuura

#### 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 $PGL(2,O)$.

#### Article information

Source
Algebra Number Theory, Volume 4, Number 3 (2010), 255-295.

Dates
Revised: 28 November 2009
Accepted: 29 November 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513729525

Digital Object Identifier
doi:10.2140/ant.2010.4.255

Mathematical Reviews number (MathSciNet)
MR2602667

Zentralblatt MATH identifier
1205.11064

#### Citation

Matsuura, Ryota. Twisted root numbers of elliptic curves semistable at primes above 2 and 3. Algebra Number Theory 4 (2010), no. 3, 255--295. doi:10.2140/ant.2010.4.255. https://projecteuclid.org/euclid.ant/1513729525

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