BD01 — Birch–Swinnerton-Dyer Conjecture
| Field | Value |
|---|---|
| Domain | Millennium Problems |
| System | Elliptic curve E/ℚ and its L-function L(s,E) |
| Group | GL_2(𝔸_ℚ) (automorphic side) / E(ℚ) ≅ ℤ^r ⊕ E(ℚ)_tors (arithmetic side) |
| H^k tier | H² |
| ISA | Meld (β→0) |
| Status | Conjectured |
| Opcodes | ORBIT · TWIST · BIND · LABEL |
| Papers | Paper 492, Paper 543, Paper 551 |
Physical system
The Birch–Swinnerton-Dyer (BSD) conjecture concerns the arithmetic of elliptic curves E: y² = x³ + ax + b over ℚ. The group of rational points E(ℚ) is a finitely generated abelian group by the Mordell-Weil theorem:
E(ℚ) ≅ ℤ^r ⊕ E(ℚ)_tors
where r ≥ 0 is the rank (number of independent infinite-order rational points) and E(ℚ)_tors is the finite torsion subgroup (at most 16 elements, Mazur’s theorem).
The L-function (from LA02) is:
L(s,E) = ∏_{p ∤ N} (1 − a_p p^{−s} + p^{1−2s})^{-1} × (local factors at bad primes)
The BSD conjecture (Birch–Swinnerton-Dyer 1965): the rank r of E(ℚ) equals the order of vanishing of L(s,E) at s=1:
rank E(ℚ) = ord_{s=1} L(s,E)
This connects a purely algebraic invariant (rank = dimension of the free part of E(ℚ)) to an analytic invariant (vanishing order of an L-function at a specific point). Neither is easy to compute; the conjecture says they are equal.
BSD refined conjecture: the leading coefficient of the Taylor expansion at s=1 equals an explicit product of arithmetic invariants (real period, regulator, Sha order, torsion):
| L^{(r)}(1,E) / r! = (Ω_E · Reg_E · ∏_p c_p · | Ш(E) | ) / | E(ℚ)_tors | ² |
Target category
SelmerCat(E/ℚ) — the category whose objects are the Selmer groups Sel_n(E/ℚ) for each integer n ≥ 1 (the H¹ Galois cohomology groups capturing global-to-local obstructions), and whose morphisms are the maps between them induced by n-isogenies. The Tate-Shafarevich group Ш(E/ℚ) = lim_n Sel_n / E(ℚ)/n is the H² content — the obstruction to the H¹ Selmer group being generated by actual rational points.
BSD = the claim that the BIND multiplicity (order of vanishing of L) equals the TWIST rank (dimension of free H¹ = rank of E(ℚ)).
Interpretation functor
F: C → SelmerCat(E/ℚ) defined by:
| Opcode | F(opcode) |
|---|---|
| ORBIT | Rational points: each P = (x,y) ∈ E(ℚ) is a ORBIT step; the group law P+Q on E is the ORBIT composition; rank r = dimension of the ORBIT lattice; generator search = ORBIT on E(ℚ)/torsion |
| TWIST | Selmer group Sel_n(E/ℚ): the H¹(Gal, E[n]) cohomology class measuring “near misses” — elements that are locally rational at every prime p but may not be globally rational; the TWIST rank is an upper bound for r |
| BIND | Tate-Shafarevich group Ш(E/ℚ): the H² obstruction measuring elements of Sel that are not actual rational points; Ш = ker(H¹ global → ∏v H¹ local); conjecturally finite; the BIND multiplicity = ord{s=1} L(s,E) if BSD holds |
| LABEL | a_p(E) = p+1−#E(𝔽_p): the LABEL at each good prime; these are the Euler product factors; the Taylor coefficient L^{(r)}(1,E)/r! is the global LABEL eigenvalue of the BIND |
ISA programme
CURVE: LABEL[E: y^2 = x^3 + ax + b | elliptic curve over Q]
RANK: ORBIT[rank r = dim free part of E(Q) | count independent rational points]
LFUN: LABEL[L(s,E) = prod_p (1 - a_p p^{-s} + p^{1-2s})^{-1} | L-function]
VANISH: BIND[ord_{s=1} L(s,E) = ? | order of zero at s=1]
SELMER: TWIST[Sel_n(E/Q) in H^1(Gal, E[n]) | upper bound for rank, H1 content]
SHA: BIND[Sha(E/Q) = ker(H^1_global -> prod H^1_local) | H2 obstruction]
BSD: BIND[rank E(Q) = ord_{s=1} L(s,E) | the conjecture = BIND = TWIST rank]
REFINED: LABEL[L^(r)(1)/r! = Omega * Reg * prod_p c_p * |Sha| / |tors|^2 | formula]
Computable output
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Rank determination: for E: y²=x³−x, rank r=0 (only finitely many rational points: (0,0),(1,0),(−1,0)); L(1,E) ≠ 0 (BSD predicts r=0 ✓ — confirmed by Coates-Wiles 1977 for CM curves). For E: y²=x³−x²−10x−10 (the “congruent number curve” for n=5): rank r=1; generator P=(−1,3); L(s,E) has a simple zero at s=1 (ord=1 ✓ — BSD verified numerically; Kolyvagin: proved that Ш is finite for rank ≤ 1).
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Kolyvagin’s theorem (1990): if L(1,E) ≠ 0 (rank 0) or L’(1,E) ≠ 0 (rank 1), then BSD is true up to the finiteness of Ш. This proves BSD for rank 0 and 1 curves — the BIND (Ш) is trivial when the TWIST rank matches the analytic rank. The remaining cases (rank ≥ 2) are open.
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Rank 2 example: E: y²+y=x³−7x+6 has rank r=3 (generators: (2,0),(−1,1),(0,2)); L(s,E) has a zero of order 3 at s=1 (verified numerically; BSD predicts r=3, and this is consistent with all numerical evidence, but not proved).
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Tate-Shafarevich group Ш: for E: y²=x³−x, Ш(E/ℚ) = 0 (trivially, since rank 0 and BSD is proved). For rank ≥ 2 curves, |Ш| can be large; e.g., Ш of the “congruent number curves” grows with n. The refined BSD formula predicts |Ш| as a rational combination of the other arithmetic invariants — this is the BIND eigenvalue.
BSD as β* snap on the β-plane
The BSD conjecture is a β-plane snap at the critical point s=1:
| L(s,E) behaviour at s=1 | BSD meaning | β-plane | ISA |
|---|---|---|---|
| L(1,E) ≠ 0 | rank 0: finitely many rational points | β* snap = simple pole avoidance | H⁰ ORBIT closes |
| L(1,E) = 0, L’(1,E) ≠ 0 | rank 1: one generator of infinite order | β* snap = simple zero, single TWIST | H¹ TWIST generator |
| L^{(r)}(1,E) = first non-zero | rank r: r independent generators | r-th order zero = r-fold BIND | H² BIND multiplicity = r |
The s=1 point is the β* snap of L(s,E): at s=1, the Euler product ∏_p (local factor) transitions from the Origami regime (s>1, convergent product) to the Meld regime (s<1, non-convergent). The vanishing order at s=1 measures how many times the ORBIT product “wants” to diverge — and each divergence corresponds to one independent rational point.
The regulator Reg_E = |det(⟨P_i, P_j⟩)|: the volume of the ORBIT lattice generated by the r independent rational points. The height pairing ⟨P,Q⟩ is the TWIST inner product on E(ℚ)/tors. BSD says L^{(r)}(1)/r! ∝ Reg_E × |Ш|, i.e., the BIND eigenvalue = ORBIT lattice volume × H² obstruction. This is the ISA “BIND = ORBIT × TWIST” composition rule at the critical point.
Connection to Riemann Hypothesis (RI01)
BSD and RH are related as GL_2 ↔ GL_1 in the Langlands programme:
- RH is about the zeros of L(s, trivial) = ζ(s) on Re(s)=1/2 (BIND events of the trivial GL_1 representation)
- BSD is about the zero at s=1 of L(s,E) for the GL_2 representation of E (the BIND event of a non-trivial automorphic form)
The GRH (Generalised RH) for L(s,E) would place all zeros on Re(s)=1/2, which is distinct from BSD (which concerns the specific zero at s=1). Both are H² BIND conjectures in the Langlands programme; BSD is the “local” statement about a specific BIND at s=1, while GRH is the “global” statement about all BINDs.
Why H² (not H¹)
- Ш is H²: the Tate-Shafarevich group is a subgroup of H²(Gal, E(ℚ̄)) in Galois cohomology — a genuinely H² object (not just H¹ Selmer). Its conjectured finiteness is an H² BIND being “removable” in the right sense.
- The vanishing order is H²: counting zeros of a meromorphic function uses the argument principle (CA02 = H² BIND in complex analysis); the order of vanishing at s=1 is a winding number around a codimension-2 point.
- The Kolyvagin Euler system: the proof of BSD for rank ≤ 1 uses Euler systems (collections of Galois cohomology classes indexed by primes) — a machinery that is intrinsically H² (it operates on H¹ Selmer via cup products into H²).
Connections to other entries
- LA02 (Taniyama-Shimura): BSD builds directly on LA02; the modularity of E (proved in LA02) is a prerequisite for the analytic theory of L(s,E); without Taniyama-Shimura, L(s,E) would not be known to have an analytic continuation
- RI01 (Riemann Hypothesis): BSD is the GL_2 analogue of RH; GRH for L(s,E) = all zeros on Re(s)=1/2; BSD is the additional claim about the zero at s=1
- Paper 551 (Adelic Atom): the adèlic L-function of ζ(s) (RI01) and L(s,E) (BD01) share the same adèlic β-plane structure; Scott correction = residue at s=1 of ζ = the GL_1 analogue of the L’(1,E) value in BSD rank-1 curves
Validation
- Birch & Swinnerton-Dyer (1965), Proc. LMS: original numerical evidence; log L(1,E) ∝ (log #E(𝔽_p)) growth consistent with rank.
- Coates & Wiles (1977): BSD proved for CM elliptic curves with rank 0.
- Kolyvagin (1990): BSD proved (modulo Ш finite) for rank ≤ 1; Euler system of Heegner points; Ш finite for these curves.
- Gross-Zagier (1986): L’(1,E) ≠ 0 iff a Heegner point is non-torsion; connects L-value derivative to rank-1 rational point.
- Numerical: BSD verified for millions of curves (Cremona database); no counterexample found; rank ≥ 2 cases consistent numerically.