Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T08:30:03.639Z Has data issue: false hasContentIssue false
  • English
  • Français

Algebraicity of L-values for elliptic curves in a false Tate curve tower

Published online by Cambridge University Press:  10 April 2007

THANASIS BOUGANIS  and
VLADIMIR DOKCHITSER
THANASIS BOUGANIS
Affiliation:
Mathematisches Institut der Universität Heidelberg, Im Neuenheimer Feld 288, D-69129 Heidelberg, Germany e-mail: bouganis@mathi.uni-heidelberg.de
VLADIMIR DOKCHITSER
Affiliation:
D.P.M.MS., University of Cambridge, Wilberforce Road, Cambridge CB3 OWB. e-mail: v.dokchitser@dpmms.cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Let E be an elliptic curve over , and τ an Artin representation over that factors through the non-abelian extension , where p is an odd prime and n, m are positive integers. We show that L(E,τ,1), the special value at s=1 of the L-function of the twist of E by τ, divided by the classical transcendental period Ω+d+|Ω−d−|ε(τ) is algebraic and Galois-equivariant, as predicted by Deligne's conjecture.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 2 , March 2007 , pp. 193 - 204
Copyright
Copyright © Cambridge Philosophical Society 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Arthur, J. and Clozel, L.. Simple algebras, base change and the advanced theory of the trace formula. Annals of Math. Studies 120 (1989).Google Scholar
[2] Birch, B. J.. Elliptic curves, a progress report. Proceedings of the 1969 Summer Institute on Number Theory (Stony Brook, New York AMS) (1971), pp. 396–400.CrossRefGoogle Scholar
[3] Breuil, C., Conrad, B., Diamond, F. and Taylor, R.. On the modularity of elliptic curves over . J. Amer. Math. Soc. 14 (2001), 843–939.CrossRefGoogle Scholar
[4] Coates, J., Fukaya, T., Kato, K., Sujatha, R. and Venjakob, O.. The GL2 main conjecture for elliptic curves without complex multiplication. Inst. Hautes Études Sci. Publ. Math. 101 (2005), 163–208.CrossRefGoogle Scholar
[5] Deligne, P.. Valeur de fonctions L et périodes d'intégrales. Automorphic forms, representations and L-function (ed. Borel, A. and Casselman, W.). Proceedings of Symposia in Pure Mathematics 33, Part 2 (American Mathematical Society, 1979), 313–346.Google Scholar
[6] Dokchitser, V.. Root numbers of non-abelian twists of elliptic curves. Proc. London Math. Soc. (3) 91 (2005), 300–324.CrossRefGoogle Scholar
[7] Langlands, R. P.. Base change for GL(2). Annals of Math. Studies 96 (1980).Google Scholar
[8] Rohrlich, D.. On L-functions of elliptic curves and cyclotomic towers. Invent. Math. 75 (1984), 404–423.Google Scholar
[9] Rohrlich, D.. L-functions and division towers. Math. Ann. 281 (1988), 611–632.CrossRefGoogle Scholar
[10] Serre, J-P.. Local Fields. GTM 67 (Springer-Verlag, 1979).CrossRefGoogle Scholar
[11] Shimura, G.. On the periods of modular forms. Math. Ann. 229 (1977), 211–221.CrossRefGoogle Scholar
[12] Shimura, G.. The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J. 45 No 3 (1978), 637–679.CrossRefGoogle Scholar
[13] Tate, J.. Number theoretic background. Automorphic forms, representations and L-function (ed Borel, A. and Casselman, W.). Proceedings of Symposia in Pure Mathematics 33, Part 2 (American Mathematical Society, 1979), 3–26.Google Scholar
[14] Taylor, R. and Wiles, A.. Ring theoretic properties of certain Hecke algebras. Annals of Math. 141 (1995), 553–572.CrossRefGoogle Scholar
[15] Wiles, A.. Modular elliptic curves and Fermat's last theorem. Annals of Math. 141 (1995), 443–551.CrossRefGoogle Scholar