Completing the computational verification program for the Birch and Swinnerton-Dyer conjectural formula for individual elliptic curves over Q: excluded primes, deferred CM and reducible cases, and the analytic rank at least 2 regime

The 2009 computational program of Grigorov–Jorza–Patrikis–Stein–Tarniţǎ verified the Birch–Swinnerton-Dyer (BSD) conjectural formula for non-CM (no complex multiplication) elliptic curves over Q of conductor at most 1000 and analytic rank 0 or 1, except possibly at odd primes dividing Tamagawa numbers or rational cyclic isogeny degrees, except in cases where the mod-p Galois representation is reducible, and not at all for analytic rank at least 2. Later work (Miller; Creutz–Miller; Miller–Stoll; Creutz) completed the verification for all rank 0/1 curves of conductor below 5000, and Stein–Wüthrich computed vanishing of p-primary Tate–Shafarevich groups for many curves of Mordell–Weil rank at least 2, while Burungale–Skinner–Tian–Wan established the first infinite families of non-CM curves for which the BSD formula is provable (still analytic rank at most 1) and Banwait–Huang algorithmically identified such curves up to conductor 500000. The open problem, faithful to the source, is to carry this verification program beyond the exclusions it left: to prove $\mathrm{BSD}(E,p)$ in the deferred reducible-representation and excluded-prime cases, to treat the CM curves the source set aside, and — the essential surviving core — to reach analytic rank at least 2, where even a single elliptic curve over Q has no verified formula, while continuing to extend proof-grade verification beyond conductor 5000.

Background

An elliptic curve $E$ over the rationals $\mathbb{Q}$ is a smooth projective curve of genus 1 with a rational point, equipped with a Hasse–Weil $L$ -function $L(E,s)$ , an analytic function built from the point counts of the reductions of $E$ modulo primes. Its order of vanishing at $s=1$ is the analytic rank. The Birch and Swinnerton-Dyer (BSD) conjecture predicts that the analytic rank equals the Mordell–Weil rank $r$ (the rank of the finitely generated abelian group $E(\mathbb{Q})$ ) and that the leading coefficient is arithmetic: with $\Omega_E$ the Néron period (the real period, obtained by integrating a Néron differential over $E(\mathbb{R})$ ; equivalently, the fundamental real period, doubled when $E(\mathbb{R})$ has two connected components, uniformly in $r$ ), $\mathrm{Reg}(E)$ the regulator (covolume of $E(\mathbb{Q})$ modulo torsion), $c_\ell(E)$ the Tamagawa numbers (indices of $E(\mathbb{Q}_\ell)$ -points of the connected component of the special fiber of the Néron model), $E(\mathbb{Q})_{\mathrm{tors}}$ the torsion subgroup, and $\mathrm{III}(E/\mathbb{Q})$ the Tate–Shafarevich group (the group measuring failure of the local–global principle for principal homogeneous spaces, conjecturally finite), the conjectural formula reads

\frac{L^{(r)}(E,1)}{r!} \;=\; \frac{\Omega_E\,\mathrm{Reg}(E)\,\prod_{\ell} c_\ell(E)\;\lvert \mathrm{III}(E/\mathbb{Q})\rvert}{\lvert E(\mathbb{Q})_{\mathrm{tors}}\rvert^{2}}.

For a prime $p$ , verifying $\mathrm{BSD}(E,p)$ means, following the source's convention (its Conjecture 1.3), proving that $\mathrm{III}(E/\mathbb{Q})[p^{\infty}]$ is finite and that $\mathrm{ord}_p$ of its order equals $\mathrm{ord}_p$ of the analytic order predicted by the formula, with the rank part of BSD (equality of algebraic and analytic ranks) assumed; for analytic rank at most 1 the rank part is a theorem, whereas for analytic rank at least 2 it — and even the rationality of the analytic order of $\mathrm{III}$ (source, Remark 1.7) — is itself open and must be established as part of any verification. By Gross–Zagier and Kolyvagin (via Heegner points — special points of $E$ defined over imaginary quadratic fields, constructed using the theory of complex multiplication — and Euler systems — families of cohomology classes in Galois cohomology whose norm relations bound Selmer groups and hence Tate–Shafarevich groups), the rank part and finiteness of $\mathrm{III}$ are theorems for analytic rank at most 1, and Kato's Euler system gives the relevant divisibility in many cases; no technique is known for analytic rank at least 2. Grigorov–Jorza–Patrikis–Stein–Tarniţǎ (2009) combined these theorems with rigorous computation of $L$ -values via modular symbols (explicit linear functionals on cohomology of modular curves that compute $L(E,1)$ and its derivative exactly up to controlled error) to verify the full formula for all non-CM curves of conductor at most 1000 and rank 0 or 1 (CM, complex multiplication, means the endomorphism ring of the curve over an algebraic closure is strictly larger than $\mathbb{Z}$ ; the computational methods and Iwasawa-theoretic inputs below require the non-CM case), except at odd primes dividing a Tamagawa number or the degree of a rational cyclic isogeny (a nonconstant rational map $E \to E'$ with cyclic kernel), and except in cases where the mod- $p$ Galois representation $\rho_{E,p}$ on the $p$ -torsion points is reducible, which breaks the irreducibility hypotheses of the underlying results from Iwasawa theory (the study of arithmetic in $\mathbb{Z}_p$ -extensions via $p$ -adic $L$ -functions and Selmer groups) and Kolyvagin-type arguments.

Problem Statement

Complete the computational-verification program of Grigorov–Jorza–Patrikis–Stein–Tarniţǎ for the Birch–Swinnerton-Dyer conjectural formula for individual elliptic curves over $\mathbb{Q}$ : verify $\mathrm{BSD}(E,p)$ in the cases the source excluded or deferred. Concretely, the open directions marked by the source and its continuation are: (1) curves of analytic rank $r \ge 2$ , where the formula has not been verified for even one elliptic curve over $\mathbb{Q}$ and where even the rationality of the analytic order of $\mathrm{III}$ is unknown — this requires proving that the algebraic rank equals the analytic rank and that $\mathrm{III}(E/\mathbb{Q})[p^{\infty}]$ is finite with the order predicted by the conjectural formula stated in the background, beyond the reach of the Gross–Zagier, Kolyvagin, and Kato theorems that only cover analytic rank at most 1; (2) the cases excluded from the source's main theorem: odd primes dividing Tamagawa numbers or the degree of a rational cyclic isogeny, and primes at which the mod- $p$ Galois representation $\rho_{E,p}$ is reducible — a case the source explicitly deferred to later theoretical work (Stein–Wüthrich) — where the standard hypotheses of the underlying Iwasawa-main-conjecture and Euler-system arguments fail; and (3) the CM elliptic curves the source excluded from most of its computations, whose treatment it described as requiring different methods for both parts of the BSD conjecture. Continuing to extend the proof-grade rank 0 and 1 verification tables beyond their current conductor bounds (1000 in the source, 5000 in later work), with rigorous error-controlled computation of $L^{(r)}(E,1)$ and certified descents, is a further direction of the same program. Verification for every elliptic curve over $\mathbb{Q}$ and every prime would amount to the BSD conjecture itself, a Clay Millennium problem; the source's program approaches that conjecture through individual curves. The named directions (1)-(3) and the table extension are independent targets rather than jointly required: a complete treatment of any single one of them settles $\mathrm{BSD}(E,p)$ in the corresponding cases and leaves the others open, with direction (1), the analytic rank $r \ge 2$ regime, its essential core.

References

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