Math Conjectures

Birch–Swinnerton-Dyer Conjecture

OPENIconicConjectureProposed 1965 · Standard version

Canonical statement

Let E/QE/\mathbb Q be an elliptic curve. Write E(Q)≅E(Q)tors⊕ZrE(\mathbb Q)\cong E(\mathbb Q)_{\mathrm{tors}}\oplus\mathbb Z^r, and let L(E,s)L(E,s) be the Hasse–Weil LL-function of EE, continued to an entire function by modularity. Then
ord⁡s=1L(E,s)=r. \operatorname{ord}_{s=1}L(E,s)=r.
View source LaTeX
Let \(E/\mathbb Q\) be an elliptic curve. Write \(E(\mathbb Q)\cong E(\mathbb Q)_{\mathrm{tors}}\oplus\mathbb Z^r\), and let \(L(E,s)\) be the Hasse–Weil \(L\)-function of \(E\), continued to an entire function by modularity. Then
\[
  \operatorname{ord}_{s=1}L(E,s)=r.
\]
NT-ARITH-002:[BirchSwinnertonDyer1965][Wiles2006BSD][GrossZagier1986]
Number theoryAlgebraic geometry

Notes

Let EE be an elliptic curve over Q\mathbb Q with Mordell–Weil rank rr, and let L(E,s)L(E,s) be its Hasse–Weil LL-function, entire by modularity. The Birch–Swinnerton-Dyer conjecture, in the standard rank form recorded here, predicts that the order of vanishing of L(E,s)L(E,s) at s=1s=1 equals rr. It grew out of the machine computations of Birch and Swinnerton-Dyer, published in 1965 [BirchSwinnertonDyer1965].

The strongest general evidence lies at low analytic rank. Gross and Zagier related the derivative L′(E,1)L'(E,1) to the height of a Heegner point [GrossZagier1986], and together with later Euler-system methods this establishes the rank equality in many analytic-rank 00 and 11 cases and in important families; see Wiles's account of the problem [Wiles2006BSD]. The refined conjecture, which expresses the leading Taylor coefficient at s=1s=1 in terms of arithmetic invariants of EE, is strictly stronger and is not part of this version.

For arbitrary curves, and in particular whenever the rank is at least 22, the equality remains open in both directions.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.