nLab Birch and Swinnerton-Dyer conjecture

Idea

The Birch and Swinnerton-Dyer conjecture is a conjecture about the form of the first non-vanishing derivative of the Hasse-Weil L-function of an elliptic curve at $s= 1$, expressed in terms of a higher regulator, analogous to the class number formula for a Dedekind zeta function.

This is hence a conjecture about special values of L-functions. It influenced the more far-reaching Beilinson conjectures.

Statement

Let $E$ be an elliptic curve over $\mathbb{Q}$. The Mordell-Weil theorem states that the Mordell-Weil group $E(\mathbb{Q})$ (defined to be the group of rational points of $E$) is a finitely generated abelian group, i.e. it is isomorphic to the product of $r$ copies of $\mathbb{Z}$ with some torsion subgroup. This $r$ is called the algebraic rank.

The rank part of the Birch and Swinnerton-Dyer conjecture states that the algebraic rank $r$ is given by the order of vanishing of the Hasse-Weil L-function $L(E,s)$ of $E$ at $s=1$. For instance, if $L(E,s)$ does not vanish at $s=1$, then the conjecture states that $E$ has no rational points of infinite order.

Note that more generally it is not known if the Hasse-Weil L-function of an elliptic curve (over a more general field) is even defined at $s=1$. However the modularity theorem states that every elliptic curve over $\mathbb{Q}$ is modular, so this Hasse-Weil L-function is equal to the L-function of some modular form, which is known to have analytic continuation to all of $\mathbb{C}$, by the work of Erich Hecke.

There is an even stronger statement of the Birch and Swinnerton-Dyer conjecture that gives the first non-vanishing Taylor coefficient of the Hasse-Weil L-function of $E$:

$$\frac{L^{(r)}(E,1)}{r!}=\frac{\#Sha(E) \Omega_{E} R_{E}\prod_{p\vert N}c_{p}}{(#E(\mathbb{Q})_{Tor})^{2}}.$$

Here

• $\#Sha(E)$ is the order of the Tate-Shafarevich group of $E$

• $\Omega_{E}$ is the real period of $E$

• $R_{E}$ is the regulator of $E$

• $c_{p}$ is the Tamagawa number of $E$ at a prime $p$ dividing the conductor $N$ of $E$

• $\#E(\mathbb{Q})_{Tor}$ is the order of the torsion subgroup of the Mordell-Weil group $E(\mathbb{Q})$ of $E$.

Progress

It is known that if the L-function does not vanish at $s=1$ then the elliptic curve has no rational points of infinite order. It is also known that if the L-function has a zero of order $1$ at $s=1$ then the rank of the elliptic curve is $1$.

These results follow from the work of Benedict Gross and Don Zagier on the Gross-Zagier formula and Heegner points, and Victor Kolyvagin?‘s work on Euler systems (which bound the Selmer group, which in turn control the rational points).

Related concepts

• Picard group, Tate-Shafarevich group

• Tamagawa number

References

• Wikipedia, Birch and Swinnerton-Dyer conjecture

• Frank Gounelas, The BSD cconjecture, regulators and special values of L-functions (pdf)

• Spencer Bloch, A note on height pairings, Tamagwawa numbers, and the Birch and Swinnerton-Dyer conjecture, Inventiones math. 58, 65-76 (1980) (pdf)

• Conrad, Venkatesh, et al., BSD Seminar Introduction to the BSD conjecture, All lecture notes