# 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} }

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…

## 51 Citations

Review of the Birch and Swinnerton-Dyer Conjecture

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- 2016

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…

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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…

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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…

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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…

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- Mathematics
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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…

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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…

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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…

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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}}
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Noninvariance of Weak Approximation Properties Under Extension of the Ground Field

- MathematicsMichigan Mathematical Journal
- 2022

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

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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…

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