The Birch and Swinnerton--Dyer conjecture famously predicts that the rank of an elliptic curve can be computed from its $L$-function. In this article we consider a weaker version called parity prove following. Let $E_1$ $E_2$ two curves defined over number field $K$ whose 2-torsion groups are isomorphic as Galois modules. Assuming finiteness Shafarevich-Tate $E_2$, show Swinnerton-Dyer correctl...

The arithmetic theory of elliptic curves enters the new century with some of its major secrets intact. Most notably, the Birch and Swinnerton-Dyer conjecture, which relates the arithmetic of an elliptic curve to the analytic behaviour of its associated L-series, is still unproved in spite of important advances in the last decades, some of which are recalled in Chapter 1. In the 1960’s the pione...

Let E be an optimal elliptic curve over Q of conductor N , such that the L-function of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N are split and such that the L-function of E over K also vanishes to order one at s = 1. In view of the Gross-Zagier theorem, the Birch and Swinnerton-Dyer conjecture says that the index in E(K) of the sub...

4 The Birch and Swinnerton-Dyer conjecture 18 4.1 Analytic rank 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.1.1 Kolyvagin’s proof . . . . . . . . . . . . . . . . . . . . . 18 4.1.2 A variant . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.1.3 Kato’s proof . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Analytic rank 1 . . . . . . . . . . . . . . . . . . . . . . . . ...

We provide two proofs that the conjecture of Artin-Tate for a fibered surface is equivalent to Birch-Swinnerton-Dyer Jacobian generic fibre. As byproduct, we obtain new proof theorem Geisser relating orders Brauer group and Tate-Shafarevich group.

We study Rubin’s variant of the p-adic Birch and Swinnerton-Dyer conjecture for CM elliptic curves concerning certain special values of the Katz two-variable p-adic L-function that lie outside the range of p-adic interpolation.

[3] , Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero, Math. Finiteness results for modular curves of genus at least 2, Amer.

Let E/Q be an optimal elliptic curve of analytic rank zero. It follows from the Birch and Swinnerton-Dyer conjecture for curves zero that order torsion subgroup divides product Shafarevich–Tate group E/Q, (global) Tamagawa number at infinity. This consequence was noticed by Agashe Stein in 2005. In this paper, we prove divisibility statement unconditionally many cases, including case where is s...

Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the padic L-function of a modular abelian variety and showing it satisfies the appropriate interpolation property. We descri...