Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication.
The annihilation of divisor classes in abelian extensions of the rational function field
The arithmetic of function fields
The average rank of elliptic curves I. With Appendix "The Explicit formula for elliptic curves over function fields" by Oisín McGuinness.
The Bloch-Kato conjecture on special values of -functions. A survey of known results
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
The Congruences of Clausen- von Staudt and Kummer for Bernoulli-Hurwitz Numbers.
The duality of cusp singularities.
The group of automorphisms of algebraic function fields.
The Hasse zeta function of a K3 surface related to the number of words of weight 5 in the Melas codes.
The local monodromy as a generalized algebraic correspondence (with an appendix by Spencer Bloch).
The "main conjectures" of Iwasawa theory for imaginary quadratic fields.
The meromorphic continuation of a zeta function of Weil and Igusa Type.
The modularity of the Barth-Nieto quintic and its relatives.
The periods of abelian varieties with complex multiplication and the spectral values of certain zeta functions
The polynomial and elliptic curves of conductor 11
The primitive conjugacy classes and the generalized homology group of a finite graph
The sign of the functional equation of the L-function of an orthogonal motive.
The work of Gross and Zagier on Heegner points and the derivatives of -series
Torsion and Tamagawa numbers
Let be a number field, and let be an abelian variety. Let denote the product of the Tamagawa numbers of , and let denote the finite torsion subgroup of . The quotient is a factor appearing in the leading term of the -function of 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 or quadratic extensions , and for abelian surfaces . The smallest possible ratio...