Birch and Swinnerton-Dyer conjecture

In mathematics, the Birch and Swinnerton-Dyer conjecture is a major unsolved problem in the field of algebraic geometry and number theory and one of the seven Millennium Prize Problems.

• The conjecture discovered jointly by Peter Swinnerton-Dyer and Bryan Birch in the early 1960s both surprised the mathematical world, and also forcefully reminded mathematicians that computations remained as important as ever in uncovering new mysteries in the ancient discipline of number theory. Although there has been some progress on their conjecture, it remains today largely unproven, and is unquestionably one of the central open problems of number theory. It also has a different flavour from most other number-theoretic conjectures in that it involves exact formulae, rather than inequalities or asymptotic questions.
  • Bryan Birch, John Coates, Jean-Louis Colliot-Thélène, and Alexei Skorobogatov (August 2019). "Peter Swinnerton-Dyer (1927–2018)". Notices of the American Mathematical Society 66 (7): 1058–1067.

• Just as Weil's conjectures were about counting solutions to equations in a situation where the number of solutions is known to be finite, the BSD conjecture concerns the simplest class of polynomial equations—elliptic curves—for which there is no simple way to decide whether the number of solutions is finite or infinite.
  • Michael Harris (30 May 2017). Mathematics without Apologies: Portrait of a Problematic Vocation. Princeton University Press. p. 27. ISBN 978-1-4008-8552-7. 

• Thanks to their conjecture, Birch and Swinnerton-Dyer are two names that (to mathematicians) are as inextricably linked as the names of Laurel and Hardy, although many have been tricked into believing that there are in fact three mathematicians behind the conjecture - Birch, Swinnerton and Dyer. Birch, with his rather bumbling manner, plays Stan Laurel to Swinnerton-Dyer's rather dour Oliver Hardy.
  • Marcus du Sautoy (31 May 2012). The Music of the Primes: Why an unsolved problem in mathematics matters. HarperCollins Publishers. p. 150. ISBN 978-0-00-737587-5. 

• This remarkable conjecture relates the behaviour of a function L at a point where it is not at present known to be defined to the order of a group Ш which is not known to be finite.
  • Tate, John (1974). "The Arithmetic of Elliptic Curves" (pdf). Inventiones mathematicae 23.