{"id":"4445ac26-b897-4702-815f-5f26e797f425","arxiv_id":"2608.11969","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A conditional p-part Birch-Swinnerton-Dyer formula is proved for analytic rank one elliptic curves over CM quadratic extensions of totally real fields, under the anticyclotomic Iwasawa main conjecture.","lead":"This paper proves a conditional p-adic version of the Birch-Swinnerton-Dyer formula for an elliptic curve over a quadratic imaginary extension of a totally real field, under the assumption that the anticyclotomic Iwasawa main conjecture holds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"For d even, a Tamagawa factor of the target formula is already assumed in Technical assumption–modularity-1.","rationale":"The reader correctly identifies the anticyclotomic Iwasawa main conjecture as a major unproved input. I agree that Conjecture 6.1 is load-bearing: without it, Proposition 6.4 and hence the final replacement of the Selmer group by the p-adic L-function fail. However, I find a sharper concern in the statement itself. For even d, Technical assumption–modularity-1 is not merely an unproved conjecture; it is an equality involving c_q(E/K), one of the Tamagawa numbers appearing in the RHS of the main theorem. The proof of Theorem 4.11 uses that assumption verbatim to place c_q into the Gross–Zagier product over D, and the proof of the main theorem then carries that product into the final BSD formula. Consequently, the paper does not independently establish the q-factor of the Tamagawa product for even d; it assumes it. This does not make the theorem false as a conditional statement, since the assumption is explicit. But it does mean the central claim should be read with the caveat that, for d even, one target factor is an input. The verdict CONDITIONAL remains appropriate; I do not see a reason to move to ACCEPT or REJECT. The paper is honest about the Iwasawa main conjecture and about a possible technical problem flagged in Remark 3.2, but the Tamagawa-factor-in-the-hypothesis issue is the most concrete place where the argument's conclusion depends on an assumption rather than on a derivation.","tokens_in":62982,"tokens_out":8674,"duration_ms":92242,"concrete_test":"Re-run the proof of Theorem 4.11 in the even-d case with Technical assumption–modularity-1 replaced by η_f(MD,1) = η_f(MD/q,q), i.e. the same congruence-number equality with the c_q factor omitted. If, as the current proof indicates, the only occurrence of c_q in the derivation of ∏_{u|D} c_u is through this hypothesis, then the conclusion loses exactly the q-factor. This confirms that the theorem proves all RHS factors except c_q, which is assumed rather than derived.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theorem's own hypothesis, Technical assumption–modularity-1 in Theorem 1.3 and §4.2.3, contains a factor that also appears on the RHS of the conclusion. For d even it assumes, up to a p-adic unit, η_f(MD,1) = η_f(MD/q,q)·c_q(E/K). In the proof of Theorem 4.11, exactly this equality is used to insert c_q into the product ∏_{u|D} c_u(E/K); the remaining factors of that product come from Ribet–Takahashi/Deines comparisons. Then §6.2 carries that product over D into the final BSD RHS. Thus, for d even, the final equality does not establish the q-th Tamagawa number; it imports it from the assumptions. The central claim is therefore weaker than 'proving the p-part of the BSD formula': when d is even, the formula is conditional on a relation that already knows one Tamagawa factor. This is not an internal inconsistency of the conditional statement, but it is a load-bearing circularity: removing or weakening that assumption would make the proof of the full RHS fail precisely at that factor.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a conditional p-part variant of the Birch and Swinnerton-Dyer formula for a semistable modular elliptic curve E over a totally real field F, after base change to a totally imaginary quadratic extension K/F with analytic rank one. Under a long list of basic and technical assumptions, including the anticyclotomic Iwasawa main conjecture, the author shows, up to a p-adic unit, that L'(E/K,1)/(Ω^cong_f Reg(E/K)) equals #Sha(E/K)[p∞] ∏_u c_u(E/K), where Ω^cong_f is the congruence period of the associated Hilbert newform. The proof assembles an anticyclotomic control theorem, an explicit Gross-Zagier formula, relations between congruence numbers and Shimura degrees, and the Liu-Zhang-Zhang p-adic Waldspurger formula. The main theorem is explicitly conditional on Conjecture 6.1 and on modularity and technical assumptions; in the even-degree case one of these assumptions already contains a Tamagawa factor.","tokens_in":63204,"tokens_out":8471,"duration_ms":89316,"significance":"If correct, the result is a substantial generalization to totally real fields of the rank-one p-part BSD work of Jetchev-Skinner-Wan, with the natural normalization by the congruence period. The paper is unusually explicit about its hypotheses and gives a detailed, self-contained account of the control theorem and the comparison of conventions across Yuan-Zhang-Zhang, Cai-Shu-Tian, and Liu-Zhang-Zhang. The main theorem is, however, a conditional statement: it depends on the unproved Iwasawa main conjecture, on modularity over F, and on an unnamed generalized Gross-Zagier-Kolyvagin theorem. Moreover, for even d, one Tamagawa factor is imported from the hypothesis rather than proved. These caveats significantly narrow the advertised unconditional content, though they do not invalidate the conditional theorem as stated.","major_comments":[{"comment":"For even d, Technical assumption-modularity-1 assumes the equality η_f(MD;1) = η_f(MD/q;q) c_q(E/K) up to a p-adic unit. In the proof of Theorem 4.11 this equality is used directly to substitute c_q(E/K) into the product ∏_{u|D} c_u(E/K), and the remaining factors of that product are then obtained from Ribet-Takahashi/Deines comparisons. Consequently, for even d the q-th Tamagawa number appearing on the right-hand side of the main theorem is not proved but is imported from the assumptions. The abstract and Theorem 1.3 should therefore be restated so that the even-d case is described as proving the formula up to the factor c_q(E/K), or the assumption should be presented as part of the conclusion rather than as a hypothesis that is independent of the BSD formula.","section":"Theorem 1.3; §4.2.3, proof of Theorem 4.11"},{"comment":"The paper uses a 'generalized Gross-Zagier-Kolyvagin theorem' to pass from analytic rank one over K to r_MW(E/K)=1 and finiteness of Sha(E/K), and this input is load-bearing for the key formula (3.1) and hence for the final theorem. No precise statement or reference is given for this theorem in the setting of arbitrary totally real fields F. If this theorem is not unconditionally available in the required generality, it must be added as an explicit assumption of Theorem 1.3; if it is a known theorem, a precise citation and statement are needed.","section":"§2.2.1.2; Proposition 3.15"},{"comment":"The final equality depends directly on Conjecture 6.1, the anticyclotomic Iwasawa main conjecture, which is stated without proof and acknowledged in Section 1.2 to be a target of future work. This is an essential step: Proposition 6.4 uses it to replace the anticyclotomic Selmer group by the special value of L_IW. The theorem is therefore a conditional theorem rather than a proof of the p-part BSD formula, and the abstract should make this dependence explicit in the displayed formula, not only in the surrounding discussion.","section":"§6.1; Proposition 6.4"}],"minor_comments":[{"comment":"The displayed formula in the theorem writes #Sha(E/K)[p^1], while the abstract and body use p∞; this typo should be corrected.","section":"Theorem 1.3"},{"comment":"There are OCR-style typos in the title and abstract: 'TOTALL Y', 'SWINNER TON-DYER', and 'quadra tic' should be cleaned up.","section":"Title and abstract"},{"comment":"The notation η_f(MD;1) is used before the reader can see that by Definition 4.14 one has η_f = η_f(N;1); an explicit identification of η_f(MD;1) with η_f in the even-d assumption would improve readability.","section":"§4.2.3"},{"comment":"The constant in the Kodaira-Spencer comparison is written as 2πi in the displayed formula but the surrounding text uses the symbol 2πi inconsistently; the notation should be unified.","section":"§5.2.5.3"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a substantial and careful assembly of difficult tools, and the conditional theorem may be of interest. The main concern is not technical execution but the framing: the even-degree case assumes a relation that already contains a Tamagawa number, and the theorem is conditional on the Iwasawa main conjecture. I would ask the editor to require the authors to foreground these points in the abstract and introduction before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is not an unconditional proof of the p-part of BSD in a new setting. It is a carefully labeled conditional theorem, and the main interest is the assembly over arbitrary totally real F. The genuinely new work is the anticyclotomic control theorem, the explicit Gross–Zagier formula with the congruence period, the Shimura degree and Ribet–Takahashi comparisons, and the specialization of the Liu–Zhang–Zhang p-adic Waldspurger formula. That assembly looks like real technical work, not just notation-pushing. The appendix comparing conventions across YZZ, CST, and LZZ is a useful addition. The paper is honest about the load-bearing Iwasawa main conjecture: Conjecture 6.1 is assumed, and Section 1.2 explicitly says removing it is future work. There are no fitted constants and no invented objects.\n\nThe soft spots are real but mostly in proportion. The whole conclusion depends on the unproved Iwasawa main conjecture; without it Proposition 6.4 does not go through. That is explicit, so it is a known condition rather than a hidden flaw. More concerning is the unnamed ‘generalized Gross–Zagier–Kolyvagin theorem’ invoked in Section 2.2.1 to pass from analytic rank one to algebraic rank one and finite Sha. That is a major external input and it is not cited. The referee should ask for a precise statement and reference. Third, the stress-test concern is valid. For even d, Technical assumption–modularity-1 assumes, up to a p-adic unit, the equality η_f(MD,1)=η_f(MD/q,q)c_q(E/K), and in Theorem 4.11 that equality is exactly what inserts c_q into the product over D. So for even degree, the q-th Tamagawa number is not proved; it is assumed. The theorem statement should say this explicitly, or the RHS should be written with that factor removed. This is not an internal contradiction, but it weakens the claim as a ‘proof’ of the full BSD RHS in the even-degree case.\n\nThe citation pattern looks solid. The paper builds on JSW17, Cai–Shu–Tian, Liu–Zhang–Zhang, Manning, Ribet–Takahashi, and Deines, and it credits them properly. The convention tables are a plus.\n\nWho is this for? People working on Iwasawa theory and BSD over number fields, especially those trying to push rank-one formulas beyond Q and imaginary quadratic fields. It deserves a serious referee. The referee should concentrate on Section 4, the role of Technical assumption–modularity-1, and the unnamed Gross–Zagier–Kolyvagin input, and the even-degree statement should be revised to expose the imported Tamagawa factor. I would send it to review with that expectation.","headline":"Conditional but genuinely broader p-part BSD formula over CM extensions of totally real fields; for even degree one Tamagawa factor comes from the hypotheses rather than the proof.","tokens_in":63686,"tokens_out":3761,"would_cite":true,"duration_ms":40231,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G40","11G05","11R80","11R42","11S40","11R23"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a p-adic Birch–Swinnerton-Dyer formula for elliptic curves of analytic rank one over totally imaginary quadratic extensions of totally real fields, up to a p-adic unit and conditional on the anticyclotomic Iwasawa main…","keywords":["Birch and Swinnerton-Dyer conjecture","totally real fields","Iwasawa theory","Gross–Zagier formula","Liu–Zhang–Zhang formula","Hilbert modular forms","Heegner points","Shafarevich–Tate group"],"falsifier":"A single example satisfying all hypotheses of Theorem 1.3 in which the p-adic valuations of the two sides differ would refute the theorem; a direct way to test the chain is to verify Conjecture 6.1 in one anticyclotomic $\\mathbb{Z}_p$-extension over a totally imaginary quadratic extension of a real quadratic field, since Proposition 6.4 collapses if the characteristic ideal and the p-adic L-function ideal differ.","tokens_in":62759,"feed_emoji":"🔢","tokens_out":9694,"duration_ms":88657,"temperature":0.7,"pith_summary":"The paper establishes a p-part Birch–Swinnerton-Dyer formula for a semistable modular elliptic curve $E$ over a totally real field $F$ after base change to a totally imaginary quadratic extension $K$, when the analytic rank of $E$ over $K$ is one. The stated equality holds up to a p-adic unit and only under a long list of assumptions, the most consequential being the anticyclotomic Iwasawa main conjecture, which is assumed without proof. A sympathetic reader would care because this moves the p-part of the BSD formula beyond $\\mathbb{Q}$ and imaginary quadratic base fields to arbitrary totally real fields, replacing the classical period by the congruence period of the associated Hilbert newform. If the assumptions are met, the derivative of the L-function at the central point, normalized by the regulator and congruence period, computes the p-power Shafarevich–Tate group times all Tamagawa numbers.","feed_headline":"p-part BSD formula holds over CM extensions of totally real fields","feed_subtitle":"Derivative L-value over K equals Sha times Tamagawa numbers up to a p-adic unit, conditional on Iwasawa theory.","key_machinery":"The carrying object is the congruence period $\\Omega^{\\mathrm{cong}}_{\\mathbf{f}}$, together with the chain of equalities linking it to Heegner-point heights and Selmer groups. The key formula (3.15) expresses the square of the index of a Heegner point in $E(K)$ as $\\#\\mathrm{Sha}(E/K)[p^\\infty]$ divided by $\\#H^1_{\\mathrm{ac}}(K,W)$, up to local error terms; the control theorem then replaces $\\#H^1_{\\mathrm{ac}}(K,W)$ by $L^{\\mathrm{IW}}_{E,K,\\phi}(1)$ times local factors, and the Liu–Zhang–Zhang formula identifies that value with a p-adic logarithm of the Heegner point. These identities are stitched together by Shimura-degree comparisons of Ribet–Takahashi type that turn ratios of congruence numbers into Tamagawa numbers.","core_discovery":"Theorem 1.3 asserts that, up to a p-adic unit, $$\\frac{L'(E/K,1)}{\\$\\Omega$^{\\mathrm{cong}}_{\\mathbf{f}} \\operatorname{Reg}(E/K)} = \\#\\mathrm{Sha}(E/K)[p^\\infty]\\prod_u c_u(E/K),$$ with $\\Omega^{\\mathrm{cong}}_{\\mathbf{f}} = (8\\pi^2)^d (f,f)_{U_0(N)} / \\eta_f$ the congruence period of the Hilbert newform $f$ attached to $E$. The proof chains three ingredients: an explicit Gross–Zagier formula expressing the derivative special value through the height of a Heegner point and the Tamagawa numbers at primes inert in $K$; an anticyclotomic control theorem plus the Iwasawa main conjecture that replaces the anticyclotomic Selmer group by the special value of a p-adic L-function; and the trivial-character Liu–Zhang–Zhang formula that cancels error terms. The equality is a variant of the classical BSD formula because the normalization uses the congruence period rather than a Néron period of $E$.","pith_inferences":["The author leaves the anticyclotomic Iwasawa main conjecture as future work; proving it for Hilbert modular forms over totally real fields would upgrade Theorem 1.3 from conditional to unconditional.","The method suggests a testable period conjecture: for these curves, the ratio of the classical Néron period to the congruence period should be a product of local factors that is a p-adic unit, which could be checked numerically for small real quadratic fields.","Because the Gross–Zagier and Waldspurger ingredients are formulated for abelian varieties of GL(2)-type parametrized by Shimura curves, a similar p-part BSD statement may hold for such abelian varieties, not only elliptic curves."],"forward_implications":["For any semistable modular $E$ over a totally real $F$ and CM extension $K/F$ satisfying the hypotheses, the p-part of the variant BSD formula holds up to a p-adic unit.","The Tamagawa numbers at primes dividing $D$ are produced by Shimura-degree comparisons of Ribet–Takahashi type, while those at primes dividing $M$ enter through the control theorem; both contributions are needed for the final product.","Over $F=\\mathbb{Q}$, the Iwasawa-theoretic p-adic L-function constructed here agrees with the previously studied anticyclotomic p-adic L-function up to a p-adic unit, so the two frameworks are compatible at the analytic level.","Since the normalization is by the congruence period of the Hilbert newform rather than a Néron period, the theorem should be read as the congruence-period variant of BSD; converting it to the classical statement is a separate period-comparison step."],"supporting_citations":[{"why":"Supplies the explicit Gross–Zagier and Waldspurger formulae from which the congruence-period Gross–Zagier formula is derived.","marker":"[CST14]"},{"why":"Provides the model anticyclotomic control theorem, the key Selmer-group index formula, and the analytic-rank-one p-part BSD strategy over Q that this paper adapts.","marker":"[JSW17]"},{"why":"Supplies the p-adic Waldspurger formula and the Liu–Zhang–Zhang p-adic L-function used for the trivial-character identity.","marker":"[LZZ18]"},{"why":"Proves the multiplicity-one result for Shimura curves that gives the p-adic comparison of congruence numbers and Shimura degrees.","marker":"[Man21]"},{"why":"Gives the Ribet–Takahashi comparison of Shimura degrees with Tamagawa numbers, iterated to identify the D-part of the final product.","marker":"[RT97]"},{"why":"Uses the modular-degree versus congruence-number comparison over Q that the paper generalizes to Hilbert modular forms and Shimura curves.","marker":"[ARS12]"},{"why":"Handles the j-part of the Shimura-degree comparison via p-adic uniformization, used to make the Tamagawa numbers p-adic units.","marker":"[Tak01]"},{"why":"Provides the base Gross–Zagier formula on Shimura curves and several convention comparisons relied on throughout.","marker":"[YZZ13]"}],"fun_headline_variants":["p-part BSD formula over CM extensions of totally real fields","Iwasawa main conjecture proves p-part BSD for CM fields","Gross–Zagier + Iwasawa yield BSD p-part over imaginary extensions","Analytic rank-one BSD p-part proven for CM extensions","p-adic BSD formula holds over CM fields via Iwasawa theory"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes, without proof, an Iwasawa main conjecture stating that the size of the anticyclotomic Selmer group is governed exactly by the special value of a p-adic L-function; if this equality fails, the final BSD formula does not follow.","fun_headline_variants_meta":{"raw":{"variants":["p-part BSD formula over CM extensions of totally real fields","Iwasawa main conjecture proves p-part BSD for CM fields","Gross–Zagier + Iwasawa yield BSD p-part over imaginary extensions","Analytic rank-one BSD p-part proven for CM extensions","p-adic BSD formula holds over CM fields via Iwasawa theory"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000544,"raw_usage":{"total_tokens":2609,"prompt_tokens":953,"completion_tokens":1656,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":569,"completion_tokens_details":{"reasoning_tokens":1566}},"tokens_in":569,"tokens_out":1656,"duration_ms":13257,"temperature":1.0,"reasoning_tokens":1566,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:21:48.005924+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A single example satisfying all hypotheses of Theorem 1.3 in which the p-adic valuations of the two sides differ would refute the theorem; a direct way to test the chain is to verify Conjecture 6.1 in one anticyclotomic $\\mathbb{Z}_p$-extension over a totally imaginary quadratic extension of a real quadratic field, since Proposition 6.4 collapses if the characteristic ideal and the p-adic L-function ideal differ.","supporting_citations":[],"review_version":1}