Tamagawa numbers for motives with (non-commutative) coefficients.
Tate-Shafarevich groups of the congruent number elliptic curves
The Analytic Rank of a Family of Jacobians of Fermat Curves
We study the family of curves , where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves . As a corollary we conclude that the jacobians of the curves with even analytic rank and those with odd analytic rank are equally distributed.
The Arithmetic Theory of Local Constants for abelian Varieties
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 boundary of the Eisenstein symbol.
The boundary of the Eisenstein symbol (Erratum).
The GL2 main conjecture for elliptic curves without complex multiplication
Let G be a compact p-adic Lie group, with no element of order p, and having a closed normal subgroup H such that G/H is isomorphic to Zp. We prove the existence of a canonical Ore set S* of non-zero divisors in the Iwasawa algebra Λ(G) of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to S*, we are able to define a characteristic element for every finitely generated Λ(G)-module M which has the property that the quotient of M by its p-primary...
The level of distribution of the special values of L-functions
The "main conjectures" of Iwasawa theory for imaginary quadratic fields.
The size of Selmer groups for the congruent number problem.
The size of Selmer groups for the congruent number problem, II.
The Tamagawa number conjecture of adjoint motives of modular forms
The work of Gross and Zagier on Heegner points and the derivatives of -series
Théorie d'Iwasawa p-adique locale et globale.
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...