Hostname: page-component-84b7d79bbc-fnpn6 Total loading time: 0 Render date: 2024-07-31T02:41:34.725Z Has data issue: false hasContentIssue false
  • English
  • Français

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: dohyeongkim@ymail.com
Get access Rights & Permissions [Opens in a new window]

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 

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]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