nLab Birch and Swinnerton-Dyer conjecture




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=1s= 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.


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

The rank part of the Birch and Swinnerton-Dyer conjecture states that the algebraic rank rr is given by the order of vanishing of the Hasse-Weil L-function L(E,s)L(E,s) of EE at s=1s=1. For instance, if L(E,s)L(E,s) does not vanish at s=1s=1, then the conjecture states that EE 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=1s=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 EE:

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


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

  • Ω E\Omega_{E} is the real period of EE

  • R ER_{E} is the regulator of EE

  • c pc_{p} is the Tamagawa number of EE at a prime pp dividing the conductor NN of EE

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


It is known that if the L-function does not vanish at s=1s=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 11 at s=1s=1 then the rank of the elliptic curve is 11.

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).


Last revised on August 4, 2023 at 16:06:35. See the history of this page for a list of all contributions to it.