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 2024[update], 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 fieldK to the behavior of the
Hasse–Weil L-functionL(E, s) of E at s = 1. More specifically, it is conjectured that the
rank of the
abelian groupE(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 (
Wiles 2006).
Mordell (1922) proved
Mordell's theorem: the group of
rational points on an elliptic curve has a finite
basis. This means that for any elliptic curve there is a finite subset of the rational points on the curve, from which all further rational points may be generated.
If the number of rational points on a curve is
infinite then some point in a finite basis must have infinite
order. The number of independent basis points with infinite order is called the
rank of the curve, and is an important
invariant property of an elliptic curve.
If the rank of an elliptic curve is 0, then the curve has only a finite number of rational points. On the other hand, if the rank of the curve is greater than 0, then the curve has an infinite number of rational points.
Although Mordell's theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve. The rank of certain elliptic curves can be calculated using numerical methods but (in the current state of knowledge) it is unknown if these methods handle all curves.
The natural definition of L(E, s) only converges for values of s in the complex plane with Re(s) > 3/2.
Helmut Hasse conjectured that L(E, s) could be extended by
analytic continuation to the whole complex plane. This conjecture was first proved by
Deuring (1941) for elliptic curves with
complex multiplication. It was subsequently shown to be true for all elliptic curves over Q, as a consequence of the
modularity theorem in 2001.
Finding rational points on a general elliptic curve is a difficult problem. Finding the points on an elliptic curve modulo a given prime p is conceptually straightforward, as there are only a finite number of possibilities to check. However, for large primes it is computationally intensive.
Initially, this was based on somewhat tenuous trends in graphical plots; this induced a measure of skepticism in
J. W. S. Cassels (Birch's Ph.D. advisor).[2] Over time the numerical evidence stacked up.
This in turn led them to make a general conjecture about the behavior of a curve's L-function L(E, s) at s = 1, namely that it would have a zero of order r at this point. This was a far-sighted conjecture for the time, given that the analytic continuation of L(E, s) was only established for curves with complex multiplication, which were also the main source of numerical examples. (NB that the
reciprocal of the L-function is from some points of view a more natural object of study; on occasion, this means that one should consider poles rather than zeroes.)
The conjecture was subsequently extended to include the prediction of the precise leading
Taylor coefficient of the L-function at s = 1. It is conjecturally given by[3]
where the quantities on the right-hand side are invariants of the curve, studied by Cassels,
Tate,
Shafarevich and others (
Wiles 2006):
At the time of the inception of the conjecture little was known, not even the well-definedness of the left side (referred to as analytic) or the right side (referred to as algebraic) of this equation.
John Tate expressed this in 1974 in a famous quote.[4]: 198
This remarkable conjecture relates the behavior of a function 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!
By the
modularity theorem proved in 2001 for elliptic curves over the left side is now known to be well-defined and the finiteness of Ш(E) is known when additionally the analytic rank is at most 1, i.e., if vanishes at most to order 1 at . In general both parts are still open as of 2024.
Current status
The Birch and Swinnerton-Dyer conjecture has been proved only in special cases:
Kolyvagin (1989) showed that a modular elliptic curve E for which L(E, 1) is not zero has rank 0, and a modular elliptic curve E for which L(E, 1) has a first-order zero at s = 1 has rank 1.
Rubin (1991) showed that for elliptic curves defined over an imaginary quadratic field K with complex multiplication by K, if the L-series of the elliptic curve was not zero at s = 1, then the p-part of the Tate–Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes p > 7.
There are currently no proofs involving curves with a rank greater than 1.
There is extensive numerical evidence for the truth of the conjecture.[5]
Consequences
Much like the
Riemann hypothesis, this conjecture has multiple consequences, including the following two:
Let n be an odd
square-free integer. Assuming the Birch and Swinnerton-Dyer conjecture, n is the area of a right triangle with rational side lengths (a
congruent number) if and only if the number of triplets of integers (x, y, z) satisfying 2x2 + y2 + 8z2 = n is twice the number of triplets satisfying 2x2 + y2 + 32z2 = n. This statement, due to
Tunnell's theorem (
Tunnell 1983), is related to the fact that n is a congruent number if and only if the elliptic curve y2 = x3 − n2x has a rational point of infinite order (thus, under the Birch and Swinnerton-Dyer conjecture, its L-function has a zero at 1). The interest in this statement is that the condition is easily verified.[6]
In a different direction, certain analytic methods allow for an estimation of the order of zero in the center of the
critical strip of families of L-functions. Admitting the BSD conjecture, these estimations correspond to information about the rank of families of elliptic curves in question. For example: suppose the
generalized Riemann hypothesis and the BSD conjecture, the average rank of curves given by y2 = x3 + ax+ b is smaller than 2.[7]
Generalizations
There is a version of this conjecture for general
abelian varieties over number fields. A version for abelian varieties over can be found as Conjecture F.4.1.6 in the book "Diophantine Geometry: An Introduction".[8]: 462
All of the terms have the same meaning as for elliptic curves, except that the square of the order of the torsion needs to be replaced by the product involving the
dual abelian variety. Elliptic curves as 1-dimensional abelian varieties are their own duals, i.e. , which simplifies the statement of the BSD conjecture. The regulator needs to be understood for the pairing between a basis for the free parts of and relative to the Poincare bundle on the product .
Deuring, Max (1941). "Die Typen der Multiplikatorenringe elliptischer Funktionenkörper". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 14 (1): 197–272.
doi:
10.1007/BF02940746.
S2CID124821516.