The Birch and Swinnerton-Dyer conjecture
Does the rank of an elliptic curve equal the order of vanishing of its L-function at s = 1?
Inferred from the community field; finer classification has not been reviewed.
Problem statement
Let be an elliptic curve over with Hasse–Weil L-function . The conjecture asserts
and, in its refined form, gives the leading Taylor coefficient of at explicitly in terms of the regulator, the Tate–Shafarevich group, the real period and the Tamagawa numbers.
Lean 4
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Background
The conjecture grew out of numerical experiments run by Bryan Birch and Peter Swinnerton-Dyer on the EDSAC computer in Cambridge in the early 1960s — one of the first conjectures in pure mathematics produced by machine computation.
Known partial results
- Coates–Wiles (1977) and Gross–Zagier (1986) combined with Kolyvagin (1988) establish the conjecture when the order of vanishing is or .
- Bhargava, Skinner and Zhang showed that a positive proportion of elliptic curves over satisfy the rank part of the conjecture.
- The finiteness of the Tate–Shafarevich group, which the refined statement presupposes, is itself open in general.
Recorded progress
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Discussion
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