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12 - Eisenstein Classes, Elliptic Soulé Elements and the ℓ-Adic Elliptic Polylogarithm

Published online by Cambridge University Press:  05 March 2015

By
Guido Kings
Edited by
John Coates ,
A. Raghuram ,
Anupam Saikia  and
R. Sujatha
Guido Kings
Affiliation:
University of Regensburg, Germany
John Coates
Affiliation:
University of Cambridge
A. Raghuram
Affiliation:
Indian Institute of Science Education and Research, Pune
Anupam Saikia
Affiliation:
Indian Institute of Technology, Guwahati
R. Sujatha
Affiliation:
University of British Columbia, Vancouver
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Summary

Abstract

In this paper we study systematically the ℓ-adic realization of the elliptic polylogarithm in the context of sheaves of Iwasawa modules. This leads to a description of the elliptic polylogarithm in terms of elliptic units. As an application we prove a precise relation between ℓ-adic Eisenstein classes and elliptic Soulé elements. This allows to give a new proof of the formula for the residue of the ℓ-adic Eisenstein classes at the cusps and the formula for the cup-product construction in [HK99], which relies only on the explicit description of elliptic units. This computation is the main input in the proof of Bloch–Kato's compatibility conjecture 6.2 needed in the proof of Tamagawa number conjecture for the Riemann zeta function.

Introduction

The purpose of this paper is twofold: on the one hand we prove a new and precise relation between ℓ-adic Eisenstein classes and elliptic Soulé elements using a description of the integral ℓ-adic elliptic polylogarithm in terms of elliptic units. On the other hand this relation will be used to give a new proof for the cup-product construction formula, which is the main result of [HK99] and is the main input in [Hu15] to obtain a proof of Bloch–Kato's compatibility conjecture 6.2. This new proof uses only elementary properties of elliptic units.

The explicit description of the integral ℓ-adic elliptic polylogarithm in terms of elliptic units was already one of the main results in the paper [Ki01]. There we used an approach via one-motives to treat the logarithm sheaf. But the main application of the ℓ-adic elliptic polylogarithm is in the context of Iwasawa theory, which makes it desirable to approach the elliptic polylogarithm systematically in this context. That such an approach is possible is already suggested in the ground-breaking paper [BL94].

In Iwasawa theory Kato, Perrin-Riou and Colmez pointed out the usefulness to work with ‘Iwasawa cohomology’, which is continuous Galois cohomology with values in an Iwasawa algebra.

Type
Chapter
Information
The Bloch–Kato Conjecture for the Riemann Zeta Function , pp. 239 - 296
Publisher: Cambridge University Press
Print publication year: 2015

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References

[BK90] Bloch, S., and Kato, K. 1990. L-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, Vol. I. Progr. Math., 86, 333–400. Birkhäuser Boston, Boston, MA.
[BL94] Beilinson, A., and Levin, A. 1994. The elliptic polylogarithm. 55, 123–190. Amer. Math. Soc.Google Scholar
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[Bl15] Blasius, D. 2015. Motivic Polylogarithm and related classes, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 193–209.
[Co98] Colmez, P. 1998. Théorie d'Iwasawa des représentations de de Rham d'un corps local. Ann. of Math., 148, no. 2, 485–571.Google Scholar
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[HK99] Huber, A., and Kings, G. 1999. Degeneration of l-adic Eisenstein classes and of the elliptic polylog. Invent. Math., 135, 545–594.Google Scholar
[HK06] Hornbostel, J., and Kings, G. 2006. On non-commutative twisting in étale and motivic cohomology. Ann. Inst. Fourier (Grenoble), 56, 1257–1279.Google Scholar
[Hu15] Huber, A. 2015. A Motivic Construction of the Soulé Deligne Classes, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 210–238.
[Ja88] Jannsen, U. 1988. Continuous étale cohomology. Math. Ann., 280, 207–245.Google Scholar
[Kt93] Kato, K. 1993. Iwasawa theory and p-adic Hodge theory. Kodai Math. J., 16, 1–31.Google Scholar
[Kt04] Kato, K. 2004. p-adic Hodge theory and values of zeta functions of modular forms. Astérisque, ix, 117–290.Google Scholar
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[So81] Soulé, C. 1981. On higher p-adic regulators. Lect. Notes in Math., 854, 372–401. Springer-Verlag.

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