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**The Tamagawa number conjecture for CM elliptic curves.**
*(English)*
Zbl 1159.11311

The Bloch-Kato conjecture describes the special values of the \(L\)-function of a smooth proper variety \(X\) over a number field \(K\) in terms of the regulator maps of the \(K\)-theory of \(X\) into Deligne cohomology and \(p\)-adic étale cohomology, respectively. The most widely used formulation of this (Tamagawa number) conjecture in Iwasawa theory is due to K. Kato [Iwasawa theory and \(p\)-adic Hodge theory, Kodai Math. J. 16, No. 1, 1–31 (1993; Zbl 0798.11050)] which is also thoroughly recalled in the paper under review. There are only two special cases for which this conjecture is proven so far in the non-critical situation, both being accomplished by S. Bloch and K. Kato in 1990.

In this article, the author provides a proof of the (weak form of) the Bloch-Kato conjecture for elliptic curves with complex multiplication by the ring of integers in a quadratic number field. His powerful approach is based on a method combining several new ingredients in very subtle and effective a manner. First, there is an explicit description of the Deligne regulator map for elliptic CM curves due to C. Deninger [Invent. Math. 96, No. 1, 1–69 (1989; Zbl 0721.14004)] using Eisenstein symbols à la Beilinson. This leads to a proof of the weak form of the Beilinson conjecture for CM elliptic curves. Secondly, A. Huber and the author have established, in an earlier paper [cf. Invent. Math., 135, No. 3, 545–594 (1999; Zbl 0955.11027)] the relation of the \(p\)-adic regulator of the Eisenstein symbol with the specialization of the \(p\)-adic elliptic polylogarithm sheaf, which now leaves the problem of computing these specializations in \(p\)-adic cohomology in order to tackle the Bloch-Kato conjecture for CM elliptic curves. Whereas the absolute Hodge realization of the elliptic polylogarithm was well understood, due to the extensive work of A. Beilinson and M. Levin (1994), a theory of the \(p\)-adic realization giving manageable étale cohomology classes in the elliptic case had to be developed as a further new ingredient. This is done by exhibiting the elliptic polylogarithm as an inverse limit of \(p^n\)-torsion points of certain one-motives, which leads then to an explicit description of the cohomology classes of the elliptic polylogarithm sheaf by classes of certain line bundles. Finally, this is combined with K. Rubin’s “Main Conjecture” in Iwasawa theory for imaginary quadratic number fields [cf. K. Rubin, Invent. Math. 103, No. 1, 25–68 (1991; Zbl 0737.11030)] to obtain a bound on the kernel and the cokernel of the Soulé map [cf. C. Soulé, Duke Math. J. 54, 249–269 (1987; Zbl 0627.14010)] from elliptic units to the étale cohomology.

At the end of the paper, all the various new results a put together and yield a complete proof of the Bloch-Kato conjecture for CM elliptic curves (Main Theorem 1.1.5).

In this article, the author provides a proof of the (weak form of) the Bloch-Kato conjecture for elliptic curves with complex multiplication by the ring of integers in a quadratic number field. His powerful approach is based on a method combining several new ingredients in very subtle and effective a manner. First, there is an explicit description of the Deligne regulator map for elliptic CM curves due to C. Deninger [Invent. Math. 96, No. 1, 1–69 (1989; Zbl 0721.14004)] using Eisenstein symbols à la Beilinson. This leads to a proof of the weak form of the Beilinson conjecture for CM elliptic curves. Secondly, A. Huber and the author have established, in an earlier paper [cf. Invent. Math., 135, No. 3, 545–594 (1999; Zbl 0955.11027)] the relation of the \(p\)-adic regulator of the Eisenstein symbol with the specialization of the \(p\)-adic elliptic polylogarithm sheaf, which now leaves the problem of computing these specializations in \(p\)-adic cohomology in order to tackle the Bloch-Kato conjecture for CM elliptic curves. Whereas the absolute Hodge realization of the elliptic polylogarithm was well understood, due to the extensive work of A. Beilinson and M. Levin (1994), a theory of the \(p\)-adic realization giving manageable étale cohomology classes in the elliptic case had to be developed as a further new ingredient. This is done by exhibiting the elliptic polylogarithm as an inverse limit of \(p^n\)-torsion points of certain one-motives, which leads then to an explicit description of the cohomology classes of the elliptic polylogarithm sheaf by classes of certain line bundles. Finally, this is combined with K. Rubin’s “Main Conjecture” in Iwasawa theory for imaginary quadratic number fields [cf. K. Rubin, Invent. Math. 103, No. 1, 25–68 (1991; Zbl 0737.11030)] to obtain a bound on the kernel and the cokernel of the Soulé map [cf. C. Soulé, Duke Math. J. 54, 249–269 (1987; Zbl 0627.14010)] from elliptic units to the étale cohomology.

At the end of the paper, all the various new results a put together and yield a complete proof of the Bloch-Kato conjecture for CM elliptic curves (Main Theorem 1.1.5).

Reviewer: Werner Kleinert (Berlin)

### MSC:

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

11G05 | Elliptic curves over global fields |

11G55 | Polylogarithms and relations with \(K\)-theory |

11G10 | Abelian varieties of dimension \(> 1\) |

11G15 | Complex multiplication and moduli of abelian varieties |

14F30 | \(p\)-adic cohomology, crystalline cohomology |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |