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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: [email protected]
VLADIMIR DOKCHITSER
Affiliation:
D.P.M.MS., University of Cambridge, Wilberforce Road, Cambridge CB3 OWB. e-mail: [email protected]
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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

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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