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A $p$-part Birch and Swinnerton-Dyer Formula over Totally Imaginary Quadratic Extension of Totally Real Fields

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2608.11969 v1 pith:U3XXEXW4 submitted 2026-08-12 math.NT

Haidong Li This is my paper · ORCID
classification math.NT MSC 11G4011G0511R8011R4211S4011R23
keywords BirchandSwinnerton-DyerconjecturetotallyrealfieldsIwasawatheoryGross–ZagierformulaLiu–Zhang–ZhangHilbertmodularformsHeegnerpointsShafarevich–Tategroup
topics Birch and Swinnerton-Dyer
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $E$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Theorem 1.3; §4.2.3, proof of Theorem 4.11] 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.
  2. [§2.2.1.2; Proposition 3.15] 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.
  3. [§6.1; Proposition 6.4] 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.
minor comments (4)
  1. [Theorem 1.3] The displayed formula in the theorem writes #Sha(E/K)[p^1], while the abstract and body use p∞; this typo should be corrected.
  2. [Title and abstract] There are OCR-style typos in the title and abstract: 'TOTALL Y', 'SWINNER TON-DYER', and 'quadra tic' should be cleaned up.
  3. [§4.2.3] 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.
  4. [§5.2.5.3] 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.

Circularity Check

1 steps flagged · score 6.0 of 10

For d even, a Tamagawa factor on the RHS of the claimed BSD formula is assumed, not derived.

  1. self definitional [Theorem 1.3, Technical assumption–modularity-1; used in the proof of Theorem 4.11 (§4.2.3) and in the final proof (§6.2).]
    "In Theorem 1.3: (Technical assumption–modularity-1) When d is even, there exists a prime ideal qjD with p ∤ Norm(q)+1 such that, up to a p-adic unit, ηf (MD; 1) = ηf (MD=q; q)·cq(E=K). In §4.2.3: Ford even,XMD,1 is not a curve. One therefore by assumption fixes a qjD and considers the curve XMD/q,q. At this point, by the second half of the first assumption of the theorem, first up to a p-adic unit one has ηf =ηf (MD=q; q)·cq(E=K)."

    The target formula in Theorem 1.3 is L'(E/K,1)/(Ω^cong_f Reg(E/K)) = #Sha(E/K)[p∞] ∏_u c_u(E/K). When d is even, the proof of the Gross–Zagier step (Theorem 4.11) inserts the q-factor of ∏_{u|D} c_u by quoting Technical assumption–modularity-1: ηf(MD,1)=ηf(MD/q,q)·cq(E/K). That equality is not a consequence of the L-value/period/regulator comparison; it is an input. Therefore the c_q(E/K) factor on the right-hand side of the final BSD equality is exactly the same quantity that the theorem assumes in its hypothesis. The derivation of that factor is tautological: the final formula contains c_q only because c_q was placed in the assumptions. The remaining factors (Sha, other c_u, regulator normalization) are still derived, so the circularity is partial and affects the d-even case.

full rationale

The paper is honest about its conditional nature: Theorem 1.3 explicitly assumes Technical assumption–modularity-1 and the anticyclotomic Iwasawa main conjecture, and §6.1 states the latter is assumed without proof. No fitted constants are hidden, and most of the derivation passes through external results (Cai–Shu–Tian, Manning, Ribet–Takahashi/Deines, Liu–Zhang–Zhang, Jetchev–Skinner–Wan). Those citations are not circular in the sense of being self-referential or of smuggling in the target formula. The genuine circularity is localized to the d-even case: a single Tamagawa factor c_q(E/K) in the conclusion is imported verbatim from Technical assumption–modularity-1. Since the theorem claims to prove the p-part BSD formula over K, the presence of one RHS factor as a hypothesis makes that factor unproved by construction. The rest of the formula—the connection between the derivative L-value, the congruence period, the regulator, Sha, and the other Tamagawa numbers—is still nontrivial and independently derived, so the paper is not wholly circular. Score 6 reflects this partial, parity-dependent reduction of the claimed result to its own input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numeric constants are fitted to data. The central claim is carried by the assumed Iwasawa main conjecture, modularity over F, an unnamed Gross-Zagier-Kolyvagin implication, and a collection of technical assumptions. There are no invented particles, forces, or new physical entities; the congruence period and the Iwasawa-theoretic p-adic L-function are normalizations and standard constructions, not new entities.

assumptions (5)
  • domain assumption Anticyclotomic Iwasawa main conjecture: char_ΛR(X_ac(M)Λ_R) = (L_IW_{E,K,phi}) as ideals.
    Stated as Conjecture 6.1 and used in Proposition 6.4; the paper explicitly says it is assumed without proof and is a target of future work.
  • domain assumption E is modular over F, meaning the associated Hilbert newform f matches the L-function, Galois representation, and Shimura curve parametrization.
    Assumed in Section 2.2.1.4 and in Theorem 1.3; modularity over general totally real fields is not proved in the paper.
  • domain assumption Analytic rank one over K implies Mordell-Weil rank one and finite Shafarevich-Tate group via a generalized Gross-Zagier-Kolyvagin theorem.
    Invoked in Section 2.2.1.2 without a cited reference; this is a substantial input over general totally real fields.
  • ad hoc to paper Technical assumptions modularity-1, modularity-2, and modularity-3 in Theorem 1.3, including the congruence-number/Tamagawa equality for even d.
    These conditions are imposed to make Theorem 4.11 and Proposition 4.25 work; they are not proved and one of them already involves a Tamagawa number.
  • domain assumption Standard analytic continuation and field assumptions: p unramified in K, F_w = Q_p, good ordinary reduction above p, E[p] irreducible over GK, and the Heegner hypothesis.
    These are listed as basic assumptions in Section 2.2.1 and Theorem 1.3.

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Pith. "Pith review of A $p$-part Birch and Swinnerton-Dyer Formula over Totally Imaginary Quadratic Extension of Totally Real Fields." pith.science (2026). https://pith.science/paper/U3XXEXW4

@misc{pith2026260811969,
  author       = {Pith},
  title        = {Pith review of: A $p$-part Birch and Swinnerton-Dyer Formula over Totally Imaginary Quadratic Extension of Totally Real Fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U3XXEXW4}},
  note         = {Machine review of arXiv:2608.11969}
}
abstract

This article studies a modular semistable elliptic curve $E$ over a totally real number field $F$ such that, upon base change to a totally imaginary quadratic extension $K$, it has analytic rank one. Assuming the Iwasawa main conjecture, along with a substantial number of assumptions, we prove a variant of the $p$-part of the Birch and Swinnerton-Dyer formula over $K$, where $p$ is an odd prime. More precisely, up to a $p$-adic unit, we have $$ \frac{L'(E/K,1)}{\Omega^{\mathrm{cong}}_{\mathbf{f}} \operatorname{Reg}(E/K)} = \# Sha(E/K)[p^\infty]\prod_{u} c_u(E/K), $$ where $\Omega^{\mathrm{cong}}_{\mathbf{f}}$ is the congruence period of the Hilbert modular form $\mathbf{f}$ associated to $E$ via the modularity conjecture.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.