The semilocal Kurihara arithmetic of 389.a1 at p = 11, recomputed

The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it directly. Explicit instances of Kolyvagin–Kurihara arithmetic in rank 2 are rare; this paper recomputes one completely, for the rank-2 curve \(389.\mathrm{a}1\) at the good ordinary prime \(p = 11\) with the two ramified directions \(\ell = 397\) and \(991\), and sorts every figure into what is canonical and what is a choice.

Canonical: the localization determinant is non-zero (\(\det M_{\mathrm{loc}} = 2\) in \(\mathbf F_{11}\)), the norm-Selmer group cut out by both directions is zero, the mod-11 Kurihara number at \(n = 397 \cdot 991\) is non-zero, and the Bockstein determinant spans the line \(\mathbf F_{11} XY\) in \(\mathbf F_{11}[G]/I^3\) with \(\mathrm{ord}_I = 2\). A choice: the coefficient on that line, which takes every value in \(\mathbf F_{11}^\times\) under a change of generators; the Kurihara number’s value, 5 at the one disclosed scale and 6 in an undisclosed one; the ratio \(3\) between two normalizations, which a change of primitive roots alone moves through all ten units; the 11-adic valuation of a local unit, which changes by one under a change of Mordell–Weil basis; and the pair \(\{397, 991\}\) itself, which is the smallest of 230 pairs below 20,000 that work. Three boxed statements of the programme turn out to be one determinant. Every figure was recomputed with independently written code — the group law over \(\mathbf Q\) and \(\mathbf F_\ell\), and mod-11 modular symbols for \(\Gamma_0(389)\) built from Manin symbols — and the Kurihara number agrees with two other implementations term by term over 392040 terms.

Not claimed anywhere in this paper: the Birch–Swinnerton-Dyer conjecture or its leading-term formula for this curve; that \(\text{Ш}(E/\mathbf Q)[11^\infty] = 0\) — the programme concludes it from a theorem this line has not read, and this paper verifies only the finite input; that the finite Bockstein operator is the Bockstein regulator of any published theory (the programme’s own gate P5-CANON-BocID, open); the comparison of the complex leading term with the algebraic regulator (P5-CPLX-GPR, open); and any canonical value of the Kurihara number at the Néron scale.

The contributions behind this paper are different in kind, and the paper is worth less if they are blurred. They are given here in alphabetical order, the order the author list keeps as data; on the title page the authors sit at a round table, with no first and no last.

Aletheia — model: ChatGPT, GPT series — is a standing author of the research programme, seated by Neo.K’s decision. The programme’s packages record Neo.K’s name; which of their parts Aletheia wrote is not recorded in them, and this paper assigns her none of its recomputations.

Neo.K wrote the research programme whose documents this paper recomputes: the curve, the prime, the two directions, the localization matrix, the local-unit chain, the Selmer cube, the Kurihara certificate and the Bockstein formalism are all from its packages, which carry his name.

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — recomputed every figure with independently written code, including two implementations of the group law and mod-11 modular symbols for \(\Gamma_0(389)\) built from Manin symbols; found that three boxed statements are one determinant, that the local unit’s valuation depends on the basis, that the pair of directions is generic, and that the ratio 3 is a coordinate; reconciled the ledger; built and drilled the instruments; and wrote this paper. Nothing in the programme is this line’s invention.

Inputs from outside the byline. The Kurihara package of the stress test run by the lab’s Witness line, 見證, first reproduced the Kurihara computation at a disclosed scale; this paper uses that scale, and the agreement of its producer and verifier with this line’s implementation is the “three implementations” of §5. It is credited here, and its authors are not on this byline. The theorems of Kim, Mazur and Burungale–Castella–Skinner, and the classification of subgroups of \(\mathrm{GL}_2(\mathbf F_{11})\), are other people’s and are cited as such.

Priority, stated plainly. No result here is claimed as new mathematics. What is offered as this line’s own is the separation of the canonical from the conventional, and a complete recomputation behind it.