On the conjecture of Birch and Swinnerton-Dyer

@article{Coates2016OnTC,
  title={On the conjecture of Birch and Swinnerton-Dyer},
  author={John Coates},
  journal={Inventiones mathematicae},
  year={2016},
  volume={39},
  pages={223-251}
}
  • J. Coates
  • Published 1 October 1977
  • Mathematics
  • Inventiones mathematicae
The conjecture of Birch and Swinnerton-Dyer is one of the principal open problems of number theory today. Since it involves exact formulae rather than asymptotic questions, it has been tested numerically more extensively than any other conjecture in the history of number theory, and the numerical results obtained have always been in perfect accord with every aspect of the conjecture. The present article is aimed at the non-expert, and gives a brief account of the history of the conjecture, its… 
Review of the Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a well known mathematics problem in the area of Elliptic Curve. One of the crowning moments is the paper by Andrew Wiles which is difficult to understand
Introduction to the Conjectures of Birch and Swinnerton-Dyer
The aim of these lectures is to introduce the Birch and Swinnerton-Dyer conjectures in its entirety. One part of these conjectures predicts the equality of two different ‘ranks’ associated to an
On the 2‐part of the Birch and Swinnerton‐Dyer conjecture for quadratic twists of elliptic curves
In the present paper, we prove, for a large class of elliptic curves defined over Q , the existence of an explicit infinite family of quadratic twists with analytic rank 0. In addition, we establish
The congruent number problem and the Birch-Swinnerton-Dyer conjecture
By introducing a new point of view in Algebraic Topology relating elliptic curves in $\mathbb{R}^2$ and suitable bordism groups, the congruent number problem is solved showing that the Tunnell's
On the Iwasawa theory of CM fields for supersingular primes
The goal of this article is two-fold: First, to prove a (two-variable) main conjecture for a CM field $F$ without assuming the $p$-ordinary hypothesis of Katz, making use of what we call the
The Birch--Swinnerton-Dyer exact formula for quadratic twists of elliptic curves
In the present short paper, we obtain a general lower bound for the 2-adic valuation of the algebraic part of the central value of the complex L-series for the quadratic twists of any elliptic curve
The split prime μ-conjecture and further topics in Iwasawa theory
The thesis is divided in three independent chapters, each focused on a different problem in Iwasawa theory. In Chapter 1 we prove the split prime μ-conjecture for abelian extensions of imaginary
Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms
This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic {\mathbb{Z}_{p}} -tower of an imaginary quadratic
Noninvariance of Weak Approximation Properties Under Extension of the Ground Field
For rational points on algebraic varieties defined over a number field $K$, we study the behavior of the property of weak approximation with Brauer-Manin obstruction under extension of the ground
The main conjecture of Iwasawa theory for elliptic curves with complex multiplication over abelian extensions at supersingular primes
Let E be an elliptic curve over an abelian extension F of an imaginary quadratic field K with complex multiplication by K. Let p be a prime number inert over K/Q (i.e. supersingular for E). We prove
...
...

References

SHOWING 1-10 OF 76 REFERENCES
Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one
We describe an algorithm to prove the Birch and Swinnerton-Dyer conjectural formula for any given elliptic curve defined over the rational numbers of analytic rank zero or one. With computer
On the Birch-Swinnerton-Dyer quotients modulo squares
Let A be an abelian variety over a number field K. An identity between the L-functions L(A/K i , s) for extensions K i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients.
Beilinson-Flach elements and Euler systems II: The Birch-Swinnerton-Dyer conjecture for Hasse-Weil-Artin -series
Let E be an elliptic curve over Q and let % be an odd, irreducible two-dimensional Artin representation. This article proves the Birch and Swinnerton-Dyer conjecture in analytic rank zero for the
On the conjectures of Birch and Swinnerton-Dyer and a geometric analog
© Association des collaborateurs de Nicolas Bourbaki, 1964-1966, tous droits reserves. L’acces aux archives du seminaire Bourbaki (http://www.bourbaki. ens.fr/) implique l’accord avec les conditions
The main conjecture for CM elliptic curves at supersingular primes
At a prime of ordinary reduction, the Iwasawa "main conjecture" for elliptic curves relates a Selmer group to a p-adic L-function. In the supersingular case, the statement of the main conjecture is
Introduction to Skinner-Urban’s Work on the Iwasawa Main Conjecture for GL2
These notes are organized as follows. In Section 2 we formulate various Iwasawa main conjectures for modular forms. We also explain an old result of Ribet to illustrate the rough idea of the strategy
Selmer groups and the indivisibility of Heegner points
For elliptic curves over Q, we prove the p-indivisibility of derived Heegner points for certain prime numbers p, as conjectured by Kolyvagin in 1991. Applications include the rened
Quadratic twists of elliptic curves
The paper generalizes, for a wide class of elliptic curves defined over Q , the celebrated classical lemma of Birch and Heegner about quadratic twists with prime discriminants, to quadratic twists by
On the modularity of elliptic curves over Q
In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E/Q is an elliptic curve, then E is modular.
...
...