Algebra & Number Theory

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

Ryota Matsuura

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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(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

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

Dates
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
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

Subjects
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]

Keywords
elliptic curves root number Mordell–Weil rank

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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References

  • S. Lang, Algebra, 3rd ed., Grad. Texts in Math. 211, Springer, New York, 2002.
  • D. E. Rohrlich, “The vanishing of certain Rankin–Selberg convolutions”, pp. 123–133 in Automorphic forms and analytic number theory (Montréal, 1989), edited by M. Ram Murty, Univ. Montréal, Montreal, QC, 1990.
  • D. E. Rohrlich, “Elliptic curves and the Weil–Deligne group”, pp. 125–157 in Elliptic curves and related topics, edited by H. Kisilevsky and M. Ram Murty, CRM Proc. Lecture Notes 4, Amer. Math. Soc., Providence, RI, 1994.
  • D. E. Rohrlich, “Galois theory, elliptic curves, and root numbers”, Compositio Math. 100:3 (1996), 311–349.
  • D. E. Rohrlich, “Root numbers of semistable elliptic curves in division towers”, Math. Res. Lett. 13:2-3 (2006), 359–376.
  • D. E. Rohrlich, “Scarcity and abundance of trivial zeros in division towers”, J. Algebraic Geom. 17:4 (2008), 643–675.
  • J.-P. Serre, “Propriétés galoisiennes des points d'ordre fini des courbes elliptiques”, Invent. Math. 15:4 (1972), 259–331.
  • J.-P. Serre, Linear representations of finite groups, Grad. Texts in Math. 42, Springer, New York, 1977.
  • A. J. Silberger, ${\rm PGL}\sb{2}$ over the $p$-adics: its representations, spherical functions, and Fourier analysis, Lecture Notes in Math. 166, Springer, Berlin, 1970. http://www.emis.de/cgi-bin/MATH-item?0204.44102Zbl 0204.44102
  • J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Grad. Texts in Math. 151, Springer, New York, 1994.