Can MathGPT solve the Birch and Swinnerton-Dyer conjecture?

No. No one has a proof, so no solver or AI can give one. MathGPT recognises the problem and shows the exact statement, its status and the established facts, labelled as reference information.

UnsolvedNumber theoryMillennium Prize Problem

Birch and Swinnerton-Dyer conjecture

Q: Is there a prize?

Prize of US \$1,000,000 offered by the Clay Mathematics Institute (Millennium Prize Problems, announced 24 May 2000).

The rank of an elliptic curve equals the order of vanishing of its L-function at s = 1.

Exact statement

E:\ y^{2}=x^{3}+ax+b,\qquad \operatorname{rank}E(\mathbb Q)=\operatorname{ord}_{s=1}L(E,s)

For an elliptic curve E over the rationals, the rank of the group of rational points E(Q) equals the order of the zero of the Hasse–Weil L-function L(E, s) at s = 1 (and a refined formula gives the leading coefficient).

Status

Open. Proposed by Bryan Birch and Peter Swinnerton-Dyer in the 1960s; proved only for analytic rank 0 and 1.

Posed: 1960s, Bryan Birch and Peter Swinnerton-Dyer (from early computer experiments).

Prize of US $1,000,000 offered by the Clay Mathematics Institute (Millennium Prize Problems, announced 24 May 2000).

In plain words

The rational points on an elliptic curve form a group, generated by finitely many points (Mordell, 1922). The rank says how many independent points of infinite order there are: rank 0 means only finitely many rational points.

The L-function is built from counting points on the curve modulo each prime. The conjecture says this analytic object, evaluated near $s=1$ , knows the rank. In particular $L(E,1)\neq0$ should mean finitely many rational points.

Key results so far

1977: Coates and Wiles: for curves with complex multiplication, L(E,1) ≠ 0 implies E(Q) is finite.
1986–1989: Gross–Zagier and Kolyvagin: if the analytic rank is 0 or 1, the rank equals the analytic rank and the Tate–Shafarevich group is finite.
2001: Modularity of all elliptic curves over Q (Wiles, Taylor–Wiles, Breuil–Conrad–Diamond–Taylor) makes L(E, s) defined everywhere.
2014: Bhargava, Skinner and Zhang: more than 66% of elliptic curves over Q (ordered by height) satisfy the rank part of the conjecture.

What our solver can and cannot do

No computer program or AI can solve the Birch and Swinnerton-Dyer conjecture by being asked. If you paste it into the MathGPT solver, you get this reference information, labelled as such, never an invented proof. The solver can still compute related special cases and examples. Open the solver.

Frequently asked questions

What does the Birch and Swinnerton-Dyer conjecture say?: For an elliptic curve E over the rationals, the rank of the group of rational points E(Q) equals the order of the zero of the Hasse–Weil L-function L(E, s) at s = 1 (and a refined formula gives the leading coefficient).
Has the Birch and Swinnerton-Dyer conjecture been solved?: Open. Proposed by Bryan Birch and Peter Swinnerton-Dyer in the 1960s; proved only for analytic rank 0 and 1.
Is there a prize?: Prize of US $1,000,000 offered by the Clay Mathematics Institute (Millennium Prize Problems, announced 24 May 2000).
Can MathGPT solve the Birch and Swinnerton-Dyer conjecture?: No. No one has a proof, so no solver or AI can give one. MathGPT recognises the problem and shows the exact statement, its status and the established facts, labelled as reference information.

References

A. Wiles, 'The Birch and Swinnerton-Dyer Conjecture', Clay Mathematics Institute (2000).
J. Silverman, The Arithmetic of Elliptic Curves, Springer (2009).

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