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This paper contains an overview of the known cases of the Bloch-Kato conjecture. It does not attempt to overview the known cases of the Beilinson conjecture and also excludes the Birch and Swinnerton-Dyer point. The paper starts with a brief review of the formulation of the general conjecture. The final part gives a brief sketch of the proofs in the known cases.

The cohomology of Monsky and Washnitzer

Marius van der Put (1986)

Mémoires de la Société Mathématique de France

The Congruences of Clausen- von Staudt and Kummer for Bernoulli-Hurwitz Numbers.

Nicholas M. Katz (1975)

Mathematische Annalen

The duality of cusp singularities.

Masa-Nori Ishida (1992)

Mathematische Annalen

The group of automorphisms of algebraic function fields.

Robert C. Valentini, Daniel J. Madden (1983)

Journal für die reine und angewandte Mathematik

The Hasse zeta function of a K3 surface related to the number of words of weight 5 in the Melas codes.

C. Peters, M. van der Vlugt, J. Top (1992)

Journal für die reine und angewandte Mathematik

The local monodromy as a generalized algebraic correspondence (with an appendix by Spencer Bloch).

Consani, Caterina (1999)

Documenta Mathematica

The "main conjectures" of Iwasawa theory for imaginary quadratic fields.

Karl Rubin (1991)

Inventiones mathematicae

The meromorphic continuation of a zeta function of Weil and Igusa Type.

D. Meuser (1986)

Inventiones mathematicae

The modularity of the Barth-Nieto quintic and its relatives.

Hulek, K., Spandaw, J., van Geemen, B., van Straten, D. (2001)

Advances in Geometry

The periods of abelian varieties with complex multiplication and the spectral values of certain zeta functions

Goro Shimura (1980)

Mémoires de la Société Mathématique de France

The polynomial $x^{3} + x^{2} + x - 1$ and elliptic curves of conductor 11

Alfred J. Van der Poorten (1976/1977)

Séminaire Delange-Pisot-Poitou. Théorie des nombres

The primitive conjugacy classes and the generalized homology group of a finite graph

А.М. Никитин (1993)

Zapiski naucnych seminarov POMI

The sign of the functional equation of the L-function of an orthogonal motive.

Takeshi Saito (1995)

Inventiones mathematicae

The work of Gross and Zagier on Heegner points and the derivatives of $L$ -series

John Coates (1984/1985)

Séminaire Bourbaki

Torsion and Tamagawa numbers

Dino Lorenzini (2011)

Annales de l’institut Fourier

Let $K$ be a number field, and let $A / K$ be an abelian variety. Let $c$ denote the product of the Tamagawa numbers of $A / K$ , and let $A {(K)}_{tors}$ denote the finite torsion subgroup of $A (K)$ . The quotient ${c / | A (K)}_{tors} |$ is a factor appearing in the leading term of the $L$ -function of $A / K$ in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise results are obtained for elliptic curves over $ℚ$ or quadratic extensions $K / ℚ$ , and for abelian surfaces $A / ℚ$ . The smallest possible ratio...

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