Conjectures
Birch and Swinnerton-Dyer Conjecture
burch and SWIN-er-tuhn DY-er kon-JEK-cher (Bryan Birch, Peter Swinnerton-Dyer)
Also Known As BSD Conjecture
Number Theory
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One of the seven Clay Mathematics Institute Millennium Prize Problems, relating two very different ways of measuring an elliptic curve's rational points: the rank of the group those points form, and the behavior at s equals one of the curve's Hasse-Weil L-function. Bryan Birch and Peter Swinnerton-Dyer found the pattern experimentally at Cambridge in the early 1960s, running early EDSAC computer calculations on elliptic curves before stating the conjecture in the form used today in 1965. The conjecture is proved in the rank zero and rank one cases, through work by John Coates and Andrew Wiles (1977), Benedict Gross and Don Zagier (1986) and Victor Kolyvagin (1989), but remains open in general.
Facts
StatementFor an elliptic curve C over the rational numbers, the order of vanishing of the L-function L(C,s) at s equals one is exactly the rank of the group of rational points on C; in particular L(C,1) equals zero if and only if that group is infinite. 1 Proposed Year Prize StatusOne of the seven Clay Mathematics Institute Millennium Prize Problems, named in 2000; a correct proof carries a one million dollar award. 1 Progress Toward ResolutionProved in the rank zero and rank one cases: Coates and Wiles (1977) showed finiteness of rational points when the curve has complex multiplication and the L-function does not vanish at one; Gross and Zagier (1986) used Heegner points to produce a point of infinite order when the L-function vanishes to order exactly one; Kolyvagin (1989 in the cited paper, described in the Clay Institute's own account as 1990) extended this to show the rank equals the order of vanishing in the zero and one cases for modular elliptic curves. The general conjecture, for higher rank, remains open. 1 The Clay Mathematics Institute's own official problem description gives Kolyvagin's result as 1990 in its prose but dates the cited underlying paper to 1989 in its own reference list; both years are carried here rather than silently resolved. Cross-Tradition Connections
In Branch
The conjecture links an arithmetic invariant (rank) to an analytic object (the L-function's behavior at a point), the same dual number-theory and analysis character the atlas already records for the Riemann Hypothesis.
Posed By
Sources
1. Clay Mathematics Institute
Clay Mathematics Institutehttps://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdfQuote, https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf
The Taylor expansion of L(C, s) at s = 1 has the form L(C, s) = c(s - 1)^r + higher order terms with c not equal to 0 and r = rank(C(Q)).
MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsPosed By: Bryan Birch, Biography section, paragraph beginning Two years later, in May 1964Quote, Posed By: Bryan Birch, Biography section, paragraph beginning Two years later, in May 1964
Two years later, in May 1964, they submitted Notes on elliptic curves II which continues their investigation and contains what today is known as the Birch-Swinnerton-Dyer Conjecture.
MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsPosed By: Peter Swinnerton-Dyer, Biography section, paragraph beginning Two years later, in May 1964Quote, Posed By: Peter Swinnerton-Dyer, Biography section, paragraph beginning Two years later, in May 1964
contains what today is known as the Birch-Swinnerton-Dyer Conjecture
Birch and Swinnerton-Dyer Conjecture (Wikipedia)
Wikimedia FoundationIn Branch: Number Theory, Background section, opening paragraphQuote, In Branch: Number Theory, Background section, opening paragraph
the Birch and 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
Open Questions (1 open question)
Does the rank of an elliptic curve's group of rational points really equal the order of vanishing of its L-function at s equals one, for every elliptic curve?
Proved only in the rank zero and rank one cases, through work by Coates and Wiles in 1977, Gross and Zagier in 1986, and Kolyvagin in 1989. No general argument is known for higher rank, and no counterexample has ever been found either.
What would resolve this A general proof covering every rank, or a single explicit elliptic curve whose rank and L-function order of vanishing disagree, reaching beyond the rank zero and rank one cases current methods can prove.
Number theoryClay Mathematics Institute
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