Diophantine geometry | Zeta and L-functions | Millennium Prize Problems | Number theory | Conjectures

Birch and Swinnerton-Dyer conjecture

In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. As of 2022, only special cases of the conjecture have been proven. The modern formulation of the conjecture relates arithmetic data associated with an elliptic curve E over a number field K to the behaviour of the Hasse–Weil L-function L(E, s) of E at s = 1. More specifically, it is conjectured that the rank of the abelian group E(K) of points of E is the order of the zero of L(E, s) at s = 1, and the first non-zero coefficient in the Taylor expansion of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K. The conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute, which has offered a $1,000,000 prize for the first correct proof. (Wikipedia).

Birch and Swinnerton-Dyer conjecture
Video thumbnail

On the Fourier coefficients of a Cohen-Eisenstein series by Srilakshmi Krishnamoorthy

12 December 2016 to 22 December 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution.

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Video thumbnail

Rigidity of p-adic local systems and Abapplications to Shimura varieties by Ruochuan Liu

12 December 2016 to 22 December 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution.

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Video thumbnail

Root numbers and parity of local Iwasawa invariants by Suman Ahmed

12 December 2016 to 22 December 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution.

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Video thumbnail

The Most Difficult Math Problem You've Never Heard Of - Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer Conjecture is a millennium prize problem, one of the famed seven placed by the Clay Mathematical Institute in the year 2000. As the only number-theoretic problem in the list apart from the Riemann Hypothesis, the BSD Conjecture has been haunting mathematicians

From playlist Math

Video thumbnail

Stark-Heegner points and generalised Kato classes by Henri Darmon

12 December 2016 to 22 December 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution.

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Video thumbnail

On the 2-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves by Zhibin Liang

12 December 2016 to 22 December 2016 VENUE Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution. An

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Video thumbnail

A twisting result in non-commutative Iwasawa theory by Somnath Jha

12 December 2016 to 22 December 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution.

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Video thumbnail

Solving Diophantine equations using elliptic curves + Introduction to SAGE by Chandrakant Aribam

12 December 2016 to 22 December 2016 VENUE Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution. An

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Video thumbnail

Solving Diophantine equations using elliptic curves + Introduction to SAGE by Chandrakant Aribam

12 December 2016 to 22 December 2016 VENUE Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution. An

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Video thumbnail

Solving Diophantine equations using elliptic curves + Introduction to SAGE by Chandrakant Aribam

12 December 2016 to 22 December 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution.

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Video thumbnail

On exceptional zero conjecture (Mazur-Tate-Teitelbaum) by Srilakshmi Krishnamoorthy

12 December 2016 to 22 December 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution.

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Video thumbnail

On Class Number of Number Fields by Debopam Chakraborty

12 December 2016 to 22 December 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution.

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Video thumbnail

Introduction to the Birch-Swinnerton-Dyer conjecture - Talk by Prof. Dr. Guido Kings (Regensburg)

Aufzeichnung des Vortrags "Introduction to the Birch-Swinnerton-Dyer conjecture" von Prof. Dr. Guido Kings (Uni Regensburg). Teil der bundesweiten Reihe "Die 7 größten Abenteuer der Mathematik" zu den Millennium-Problemen. Teil des Colloquium Wilhelm Killing der WWU Münster. Abstract: A k

From playlist Mathematics Münster News

Video thumbnail

Torsion points of the Jacobian of modular curves X0(p2 ) and non- by Debargha Banerjee

12 December 2016 to 22 December 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution.

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Video thumbnail

p-adic uniformization of locally symmetric spaces by Aditya Karnataki

12 December 2016 to 22 December 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution.

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Video thumbnail

Coates-Wiles Theorem by Anupam Saikia

12 December 2016 to 22 December 2016 VENUE Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution. An

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Video thumbnail

Comparing the corank of fine Selmer group and Selmer group of elliptic curves by Sudhanshu Shekhar

12 December 2016 to 22 December 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution.

From playlist Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture

Related pages

Order (group theory) | Tamagawa number | Congruent number | Invariant (mathematics) | Néron–Tate height | Euler product | Riemann hypothesis | Infinite set | Helmut Hasse | Tate–Shafarevich group | Complex multiplication | Generating set of a group | Main conjecture of Iwasawa theory | Millennium Prize Problems | Generalized Riemann hypothesis | Modular elliptic curve | Tate's algorithm | John Tate (mathematician) | Mathematics | Riemann zeta function | Igor Shafarevich | Abelian extension | Number theory | Rational point | Torsion group | J. W. S. Cassels | Analytic continuation | Prime number | Modularity theorem | Quadratic form | Elliptic curve | Rank of an abelian group | Tunnell's theorem | Abelian group