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p-adic L-functions over the false Tate curve extensions

Published online by Cambridge University Press:  22 July 2013

DOHYEONG KIM
DOHYEONG KIM*
Affiliation:
Department of Mathematics, POSTECH, Republic of Korea, 790-784. e-mail: [email protected]
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Abstract

Let f be a primitive modular form of CM type of weight k and level Γ0(N). Let p be an odd prime which does not divide N, and for which f is ordinary. Our aim is to p-adically interpolate suitably normalized versions of the critical values L(f, ρχ,n), where n=1,2,. . .,k − 1, ρ is a fixed self-dual Artin representation of M∞ defined by (1.1) below, and χ runs over the irreducible Artin representations of the Galois group of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. As an application, if k ≥ 4, we will show that there are only finitely many χ such that L(f, ρχ,k/2)=0, generalizing a result of David Rohrlich. Also, we conditionally establish a congruence predicted by non-commutative Iwasawa theory and give numerical evidence for it.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 3 , November 2013 , pp. 483 - 498
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Bouganis, T.Non-abelian congruences between special values of L-functions of elliptic curves; the CM case. Int. J. Number Theory 7 (2011), no. 7, 1883–1934.CrossRefGoogle Scholar
[2]Bouganis, T. and Dokchitser, V.Algebraicity of L-values for elliptic curves in a false Tate curve tower. Math. Proc. Camb. Phil. Soc. 142 (2007), no. 2, 193–204.CrossRefGoogle Scholar
[3]Coates, J., Fukaya, T., Kato, K., Venjakob, O., and Sujatha, R.The GL 2-main conjecture for elliptic curves without complex multiplication. Publ. Math. Inst. Hautes Études Sci. No. 101 (2005), 163–208.CrossRefGoogle Scholar
[4]Coates, J., Dokchitser, T., Liang, Z., Stein, W. and Sujatha, R. Non-commutative Iwasawa theory for modular forms. in preparation.Google Scholar
[5]Dokchitser, T. and Dokchitser, V.Computations in non-commutative Iwasawa theory, with V. Dokchitser and appendix by J. Coates and R. Sujatha. Proc. London Math. Soc. (3) 94 (2006), 211–272.CrossRefGoogle Scholar
[6]Delbourgo, D. and Ward, T.Non-abelian congruences between L-values of elliptic curves (Congruences non-abeliennes entres les valeurs L des courbes elliptiques). Ann. Inst. Fourier (Grenoble) 58 (2008), no. 3, 1023–1055.CrossRefGoogle Scholar
[7]Delbourgo, D. and Ward, T.The growth of CM periods over false Tate extensions. Experiment. Math. 19 (2010), no. 2, 195–210.CrossRefGoogle Scholar
[8]Gross, B.H.Arithmetic on elliptic curves with complex multiplication. Lecture Notes in Math. vol. 776 (Springer, Berlin, 1980).Google Scholar
[9]Hida, H.The Iwasawa μ-invariant of p-adic Hecke L-functions. Ann. Math. 172 (2010), 41–137.CrossRefGoogle Scholar
[10]Hida, H. and Tilouine, J.Anti-cyclotomic Katz p-adic L-functions and congruence modules. Ann. Sci. Norm. Sup. 26 (1993), 189–259.CrossRefGoogle Scholar
[11]Kato, K.K 1 of some non-commutative completed group rings. K-Theory 34 (2005), no. 2, 99–140.CrossRefGoogle Scholar
[12]Miyake, T.Modular forms. Springer Monogr. in Math. (Springer-Verlag, Berlin, 2006).Google Scholar
[13]Katz, N.p-adic L-functions for CM fields. Invent. Math. 49 (1978), no. 3, 199–297.CrossRefGoogle Scholar
[14]Rohrlich, D.On L-functions of elliptic curves and cyclotomic towers. Invent. Math. 75 (1984), no. 3, 409–423.CrossRefGoogle Scholar