Birch and Swinnerton-Dyer conjecture

History

In the early 1960s, Swinnerton-Dyer used the EDSAC-2 computer at the University of Cambridge Computer Laboratory to calculate the number ${\displaystyle N_p}$ of points modulo ${\displaystyle p}$ for a large number of primes ${\displaystyle p}$ on elliptic curves whose rank was known. From these numerical results, he and his colleague Bryan John Birch conjectured that for a curve ${\displaystyle E}$ with rank ${\displaystyle r}$, the growth of ${\displaystyle N_p}$ obeys the asymptotic law

${\displaystyle \prod _{p\le x}\frac{N_p}{p}\approx C\log (x{)}^r\text{ as }x\to \mathrm{\infty }}$,

where ${\displaystyle C}$ is a constant.

Initially, this was based on somewhat tenuous trends in graphical plots, which aroused a measure of skepticism in Birch's advisor J. W. S. Cassels. Over time, however, the accumulated numerical evidence became convincing.

This in turn led Birch and Swinnerton-Dyer to make a general conjecture about the behavior of a curve's L-function ${\displaystyle L(E,s)}$ at ${\displaystyle s=1}$; namely, that it would have a zero of order ${\displaystyle r}$ at this point. This was a far-sighted conjecture for the time, given that the analytic continuation of ${\displaystyle L(E,s)}$ was only established for curves with complex multiplication, which were also the main source of numerical examples. (Note that the reciprocal of the L-function is from some points of view a more natural object of study; using this convention, one should consider poles rather than zeroes.)

The conjecture was subsequently extended to include the prediction of the leading Taylor coefficient of the L-function at ${\displaystyle s=1}$:

${\displaystyle \frac{L^{(r)}(E,1)}{r!}=\frac{\mathrm{\#}\mathrm{S}\mathrm{h}\mathrm{a}(E){\mathrm{\Omega }}_E{R}_E\prod _{p|N}{c}_p}{(\mathrm{\#}{E}_{\mathrm{t}\mathrm{o}\mathrm{r}}{)}^2}}$,

where ${\displaystyle \mathrm{\#}{E}_{\mathrm{t}\mathrm{o}\mathrm{r}}}$ is the order of the torsion group, ${\displaystyle \mathrm{\#}\mathrm{S}\mathrm{h}\mathrm{a}(E)=\mathrm{\#}}$Ш${\displaystyle (E)}$ is the order of the Tate–Shafarevich group, ${\displaystyle {\mathrm{\Omega }}_E}$ is the real period of ${\displaystyle E}$ multiplied by the number of connected components of ${\displaystyle E}$, ${\displaystyle R_E}$ is the regulator of ${\displaystyle E}$ (defined via the canonical heights of a basis of rational points), and ${\displaystyle c_p}$ is the Tamagawa number of ${\displaystyle E}$ at a prime ${\displaystyle p}$ dividing the conductor ${\displaystyle N}$ of ${\displaystyle E}$ (computed by Tate's algorithm).

When the conjecture was originally made, little was known, not even whether the left (analytic) side or the right (algebraic) side of this equation were even well-defined. John Tate expressed this in 1974 in a famous quote:

This remarkable conjecture relates the behavior of a function ${\displaystyle L}$ 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!

By the modularity theorem proved in 2001 for elliptic curves over ${\displaystyle \mathbb{Q}}$, the left side is now known to be well-defined and the finiteness of Ш${\displaystyle (E)}$ is known when additionally the analytic rank is at most 1; i.e., if ${\displaystyle L(E,s)}$ vanishes at most to order 1 at ${\displaystyle s=1}$. For an elliptic curve over a general number field, the finiteness of both sides remains open.

In 2000, the Clay Mathematics Institute listed the Birch and Swinnerton-Dyer conjecture as one of its seven Millennium Prize Problems. The formulation of the conjecture as a prize problem was provided by Andrew Wiles, who explained its importance as follows:

Current status

The Birch and Swinnerton-Dyer conjecture has been proved in special cases:

1. Coates and Wiles proved that if ${\displaystyle E}$ is a curve over a number field ${\displaystyle F}$ with complex multiplication by an imaginary quadratic field ${\displaystyle K}$ of class number ${\displaystyle 1}$, ${\displaystyle F=K}$ or ${\displaystyle \mathbb{Q}}$, and ${\displaystyle L(E,s)}$ is not zero, then ${\displaystyle E(F)}$ is a finite group. This was extended to the case where ${\displaystyle F}$ is any finite abelian extension of ${\displaystyle K}$ by Arthaud..
2. Gross and Zagier showed that if a modular elliptic curve has a first-order zero at ${\displaystyle s=1}$, then it has a rational point of infinite order; see Gross–Zagier theorem.
3. Kolyvagin showed that a modular elliptic curve ${\displaystyle E}$ for which ${\displaystyle L(E,1)}$ is not zero has rank 0, and a modular elliptic curve ${\displaystyle E}$ for which ${\displaystyle L(E,1)}$ has a first-order zero at ${\displaystyle s=1}$ has rank 1.
4. Rubin showed that for elliptic curves defined over an imaginary quadratic field ${\displaystyle K}$ with complex multiplication by ${\displaystyle K}$, if the L-series of the elliptic curve was not zero at ${\displaystyle s=1}$, then the ${\displaystyle p}$-part of the Tate–Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes ${\displaystyle p>7}$.
5. Breuil et al. (2001), extending the work of Wiles, proved that all elliptic curves defined over the rational numbers are modular, which extends results #2 and #3 to all elliptic curves over the rationals, and shows that the L-functions of all elliptic curves over ${\displaystyle \mathbb{Q}}$ are defined at ${\displaystyle s=1}$.
6. Bhargava and Shankar proved that the average rank of the Mordell–Weil group of an elliptic curve over ${\displaystyle \mathbb{Q}}$ is bounded above by ${\displaystyle 7/6}$. Combining this with the ${\displaystyle p}$-parity theorem of Nekovář and the work of Dokchitser and Dokchitser, and with the proof of the main conjecture of Iwasawa theory for ${\displaystyle \mathrm{GL}(2)}$ by Skinner and Urban, they conclude that a positive proportion of elliptic curves over ${\displaystyle \mathbb{Q}}$ have analytic rank zero, and hence, by result #3, satisfy the Birch and Swinnerton-Dyer conjecture.

There are currently no proofs involving curves with a rank greater than 1.

There is extensive numerical evidence for the truth of the conjecture.

Consequences

Much like the Riemann hypothesis, this conjecture has multiple consequences, including:

• 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.
• 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.
• Because of the existence of the functional equation of the L-function of an elliptic curve, BSD allows us to calculate the parity of the rank of an elliptic curve. This is a conjecture in its own right called the parity conjecture, and it relates the parity of the rank of an elliptic curve to its global root number. This leads to many explicit arithmetic phenomena which are yet to be proved unconditionally. For instance:
  • Every positive integer n ≡ 5, 6 or 7 (mod 8) is a congruent number.
  • The elliptic curve given by y² = x³ + ax + b where a ≡ b (mod 2) has infinitely many solutions over ${\displaystyle \mathbb{Q}({\zeta }_8)}$.
  • Every positive rational number d can be written in the form d = s²(t³ – 91t – 182) for s and t in ${\displaystyle \mathbb{Q}}$.
  • For every rational number t, the elliptic curve given by y² = x(x² – 49(1 + t⁴)²) has rank at least 1.
  • There are many more examples for elliptic curves over number fields.

Generalizations

There is a version of this conjecture for general abelian varieties over number fields. A version for abelian varieties over ${\displaystyle \mathbb{Q}}$ is the following:

${\displaystyle \underset{s\to 1}{lim}\frac{L(A/\mathbb{Q},s)}{(s-1{)}^r}=\frac{\mathrm{\#}\mathrm{S}\mathrm{h}\mathrm{a}(A){\mathrm{\Omega }}_A{R}_A\prod _{p|N}{c}_p}{\mathrm{\#}A(\mathbb{Q}{)}_{\text{tors}}\cdot \mathrm{\#}\hat{A}(\mathbb{Q}{)}_{\text{tors}}}.}$

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 ${\displaystyle \mathrm{\#}A(\mathbb{Q}{)}_{\text{tors}}\cdot \mathrm{\#}\hat{A}(\mathbb{Q}{)}_{\text{tors}}}$ involving the dual abelian variety ${\displaystyle \hat{A}}$. Elliptic curves as 1-dimensional abelian varieties are their own duals, i.e. ${\displaystyle \hat{E}=E}$, which simplifies the statement of the BSD conjecture. The regulator ${\displaystyle R_A}$ needs to be understood for the pairing between a basis for the free parts of ${\displaystyle A(\mathbb{Q})}$ and ${\displaystyle \hat{A}(\mathbb{Q})}$ relative to the Poincaré bundle on the product ${\displaystyle A\times \hat{A}}$.

The rank-one Birch-Swinnerton-Dyer conjecture for modular elliptic curves and modular abelian varieties of GL(2)-type over totally real number fields was proved by Shou-Wu Zhang in 2001.

Another generalization is given by the Bloch-Kato conjecture.