Birch and Swinnerton-Dyer conjecture

Template:Short description Template:Use dmy dates Template:Millennium Problems 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 Sir Peter Swinnerton-Dyer, who formulated the conjecture in the 1960s with the help of machine computation.Template:Sfnp Only special cases of the conjecture have been proven.

The conjecture proposes a link between arithmetic data associated with an elliptic curve $E$ over a number field $K$ and the behaviour of its associated Hasse–Weil L-function $L(E,s)$ at $s=1$. The points $E(K)$ over a number field, for example over the rationals $\mathbb{Q}$, are subtle and difficult to determine, but the L-function depends only on the points modulo all primes $p$ and is much simpler: thus the conjecture is a local-to-global statement. 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 further that the leading non-zero coefficient in the Taylor expansion of $L(E,s)$ at $s=1$ can be expressed in terms of more refined arithmetic data attached to $E$ over $K$.Template:Sfnp

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.

Background

In 1922, Louis J. Mordell proved Mordell's theorem, stating that 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. Thus, rank zero means the curve has only a finite number of rational points, and positive rank means it 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 a given curve. The rank can sometimes be calculated using numerical methods, but it is currently unknown if these methods are effective for all curves.

An $L$-function $L(E,s)$ can be defined for an elliptic curve $E$ by constructing an Euler product from the number of points on the curve modulo each prime $p$. This $L$-function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form. It is a special case of a Hasse–Weil L-function.

The natural definition of $L(E,s)$ only converges for values of $s$ in the complex plane with $\mathrm{Re}(s)>3/2$. Helmut Hasse conjectured that $L(E,s)$ could be extended by analytic continuation to the whole complex plane.Template:Citation needed This conjecture was first proved by Max Deuring for elliptic curves with complex multiplication.Template:Sfnp It was subsequently shown to be true for all elliptic curves over $\mathbb{Q}$, as a consequence of the modularity theorem in 2001.Template:Citation needed

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, though this may be computationally intensive.

History

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

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

where $C$ 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.Template:Sfnp 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 $L(E,s)$ 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. (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 $s=1$:

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

where $\mathrm{\#}{E}_{\mathrm{t}\mathrm{o}\mathrm{r}}$ is the order of the torsion group, $\mathrm{\#}\mathrm{S}\mathrm{h}\mathrm{a}(E)=\mathrm{\#}$Template:Math$(E)$ is the order of the Tate–Shafarevich group, ${\mathrm{\Omega }}_E$ is the real period of $E$ multiplied by the number of connected components of $E$, $R_E$ is the regulator of $E$ (defined via the canonical heights of a basis of rational points), and $c_p$ is the Tamagawa number of $E$ at a prime $p$ dividing the conductor $N$ of $E$ (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:Template:Sfnp

This remarkable conjecture relates the behavior of a function $L$ at a point where it is not at present known to be defined to the order of a group Template:Math which is not known to be finite!

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

Current status

A plot, in blue, of ${\prod }_{p\le X}{N}_p/p$ for the curve $y^2=x^3-5x$ as $X$ varies over the first 100000 primes. The $X$-axis is in log(log) scale ($X$ is drawn at distance proportional to $\log (\log (X))$ from 0) and the $Y$-axis is in a logarithmic scale, so the conjecture predicts that the data should tend to a line of slope equal to the rank of the curve, which is 1 in this case; that is,  :$\frac{\log \left(\prod _{p\le X}\frac{N_p}{p}\right)}{\log C+r\log (\log X))}\to 1$ as $X\to \mathrm{\infty }$, with $C$, $r$ as in the text. For comparison, a line of slope 1 in (log(log),log)-scale with equation $\log y=a+\log (\log x)$ is drawn in red in the plot.

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

1. Coates and WilesTemplate:Sfnp proved that if $E$ is a curve over a number field $F$ with complex multiplication by an imaginary quadratic field $K$ of class number $1$, $F=K$ or $\mathbb{Q}$, and $L(E,s)$ is not zero, then $E(F)$ is a finite group. This was extended to the case where $F$ is any finite abelian extension of $K$ by Arthaud.Template:Sfnp.
2. Gross and ZagierTemplate:Sfnp showed that if a modular elliptic curve has a first-order zero at $s=1$, then it has a rational point of infinite order; see Gross–Zagier theorem.
3. KolyvaginTemplate:Sfnp 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.
4. RubinTemplate:Sfnp 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 $p>7$.
5. Template:Harvtxt, extending the work of Wiles,Template:Sfnp 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 $\mathbb{Q}$ are defined at $s=1$.
6. Bhargava and ShankarTemplate:Sfnp proved that the average rank of the Mordell–Weil group of an elliptic curve over $\mathbb{Q}$ is bounded above by $7/6$. Combining this with the $p$-parity theorem of NekovářTemplate:Sfnp and the work of Dokchitser and DokchitserTemplate:Sfnp, and with the proof of the main conjecture of Iwasawa theory for $\mathrm{GL}(2)$ by Skinner and UrbanTemplate:Sfnp, they conclude that a positive proportion of elliptic curves over $\mathbb{Q}$ 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 Template:Mvar be an odd square-free integer. Assuming the Birch and Swinnerton-Dyer conjecture, Template:Mvar is the area of a right triangle with rational side lengths (a congruent number) if and only if the number of triplets of integers (Template:Mvar, Template:Mvar, Template:Mvar) satisfying Template:Math is twice the number of triplets satisfying Template:Math. This statement, due to Tunnell's theorem Template:Harv, is related to the fact that n is a congruent number if and only if the elliptic curve Template:Math has a rational point of infinite order (thus, under the Birch and Swinnerton-Dyer conjecture, its Template:Mvar-function has a zero at Template:Math). 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 Template:Math is smaller than Template:Math.
• 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 Template:Math is a congruent number.
  • The elliptic curve given by Template:Math where Template:Math has infinitely many solutions over $\mathbb{Q}({\zeta }_8)$.
  • Every positive rational number Template:Mvar can be written in the form Template:Math for Template:Mvar and Template:Mvar in $\mathbb{Q}$.
  • For every rational number Template:Mvar, the elliptic curve given by Template:Math has rank at least Template:Math.
  • 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 $\mathbb{Q}$ is the following:Template:Sfnp

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

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

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.

References

• Template:Cite journal

• Template:Cite journal

• Template:Cite journal

• Template:Cite journal

• Template:Cite book

• Template:Cite journal

• Template:Cite journal

• Template:Cite journal

• Template:Cite journal

• Template:Cite book

• Template:Cite journal

• Template:Cite journal

• Template:Cite journal

• Template:Cite journal

• Template:Cite book

• Template:Cite journal

• Template:Cite journal

• Template:Cite journal

• Template:Cite encyclopedia

External links

• Template:MathWorld
• Template:Planetmath reference
• The Birch and Swinnerton-Dyer Conjecture: An Interview with Professor Henri Darmon by Agnes F. Beaudry
• What is the Birch and Swinnerton-Dyer Conjecture? lecture by Manjul Bhargava (September 2016) given during the Clay Research Conference held at the University of Oxford