The Tamagawa number conjecture for CM elliptic curves

By G. Kings. 2000

52 citations recorded by the source corpus.

In this paper we prove the Tamagawa number conjecture of Bloch and Kato for CM elliptic curves using a new explicit description of the specialization of the elliptic polylogarithm. The Tamagawa number conjecture describes the special values of the L-function of a CM elliptic curve in terms of the regulator maps of the K-theory of the variety into Deligne and etale cohomology. The regulator map to Deligne cohomology was computed by Deninger with the help of the Eisenstein symbol. For the Tamagawa number conjecture one needs an understanding of the $p$-adic regulator on the subspace of K-theory defined by the Eisenstein symbol. This is accomplished by giving a new explicit computation of the specialization of the elliptic polylogarithm sheaf. It turns out that this sheaf is an inverse limit of $p^r$-torsion points of a certain one-motive. The cohomology classes of the elliptic polylogarithm sheaf can then be described by classes of sections of certain line bundles. These sections are elliptic units and going carefully through the construction one finds an analog of the elliptic Soul\'e elements. Finally Rubin's ``main conjecture'' of Iwasawa theory is used to compare these elements with etale cohomology.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ed44cb884dd6765487eefdb1385ccbe94fef9adf

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In Algebra and Number Theory

1. M. Flach
2. Li Guo
3. Cornelius Greither
4. Fred Diamond
5. Kurusch Ebrahimi-Fard
6. William F. Keigher
7. Yuqun Chen
8. Leonid A. Bokut

In Geometry and Topology

1. M. Flach
2. Cornelius Greither
3. Li Guo
4. Fred Diamond
5. Werner Bley
6. Kurusch Ebrahimi-Fard
7. William F. Keigher
8. Jim Brown

In Mathematical Physics

1. M. Flach
2. Cornelius Greither
3. Li Guo
4. Fred Diamond
5. Werner Bley
6. David Burns
7. Jim Brown
8. Loïc Foissy

Citations in the corpus

• The Equivariant Tamagawa Number Conjecture: A survey (2003, background): “To illustrate we quote the main result from [53] corresponding to k < −1, j = k + 1.”
• THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES OF MODULAR FORMS (2004, result): “There are also partial results for L-functions of other modular forms at the central critical value [45,56,59,64,89] and for values of certain Hecke L-functions [49,48,57].”