Algebra & Number Theory
- Algebra Number Theory
- Volume 4, Number 3 (2010), 255-295.
Twisted root numbers of elliptic curves semistable at primes above 2 and 3
Let be an elliptic curve over a number field , and fix a rational prime . Put , where is the group of -power torsion points of . Let be an irreducible self-dual complex representation of . With certain assumptions on and , we give explicit formulas for the root number . We use these root numbers to study the growth of the rank of in its own division tower and also to count the trivial zeros of the -function of . Moreover, our assumptions ensure that the -division tower of is nonabelian.
In the process of computing the root number, we also study the irreducible self-dual complex representations of , where is the ring of integers of a finite extension of , for an odd prime. Among all such representations, those that factor through have been analyzed in detail in existing literature. We give a complete description of those irreducible self-dual complex representations of that do not factor through .
Algebra Number Theory, Volume 4, Number 3 (2010), 255-295.
Received: 15 January 2009
Revised: 28 November 2009
Accepted: 29 November 2009
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 11F80: Galois representations 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
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