Algebra & Number Theory

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

Ryota Matsuura

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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).

Article information

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

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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]

elliptic curves root number Mordell–Weil rank


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.

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