REVIEW
Elliptic units, derived Heegner classes and the p-adic Birch and Swinnerton-Dyer quotient of CM elliptic curves of rank two
Number Theory · Submitted 6 Nov 2026 · 54 references
Pith's verdict: The exact Katz-measure identification of q_p is the real result, backed by strong numerics; the square-class and derived-Heegner theorems are honest but gated by an auxiliary field whose existence is unproved, and that is what a referee should press on.
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Referee report
The paper
The paper claims
For rank-two CM curves $y^2=x^3-Dx$, the $p$-adic BSD quotient $q_p$ is the exact constant of the quadratic term of the Katz measure and the coordinate of two global classes, making $q_p=q_\infty$ testable to any precision.
The reading
What carries the argument
The load-bearing object is the two-variable Katz measure $\mu_\psi$ on $G_\infty=\operatorname{Gal}(K(fp^\infty)/K)$, $K=\mathbb{Q}(i)$: a $p$-adic measure interpolating critical Hecke $L$-values, whose degree-two term against Perrin-Riou's pencil of $p$-adic height pairings is the explicit quadratic form $(a-b)^2\det H-(a+b)^2j^2$; the exact constant of that form is $q_p$. The second mechanism is the derived Heegner class: the $\Lambda$-adic Heegner class over the anticyclotomic tower of an auxiliary imaginary quadratic field $F$ vanishes at the origin, and its derivative $D=\zeta Q_v$ is an element of $E(\mathbb{Q})\otimes\mathbb{Q}_p$ whose coordinate $\zeta$ is tied to $q_p$ by an exact identity assembled from the $\Lambda$-adic Gross-Zagier theorem, the main conjecture for CM fields, the determinant of a Selmer complex, and explicit reciprocity laws for the Bertolini-Darmon-Prasanna $p$-adic $L$-function. On the cyclotomic side, Rubin's elliptic-unit class $x^*$ plays the same role: its position in $E(\mathbb{Q})\otimes\mathbb{Q}_p$ is $q_p$ times an explicit vector, the rank-two analogue of the rank-one formula in which a single global point computes both sides.
Extended reading notes
Core claim
Assuming $\Sha(E/\mathbb{Q}(i))[p^\infty]$ is finite and the cyclotomic $p$-adic regulator $\det H$ is non-zero, Theorem 3.5 shows that the quadratic term of the two-variable Katz measure at $\psi_E$ is the explicit form $\frac14\,\iota_p(f_0\omega_E)\,\Omega_p\,e^2(1-\alpha^{-1})^2\,q_p\,[(a-b)^2\det H-(a+b)^2j^2]$, so $q_p$ is its exact constant; the same constant governs the cyclotomic, anticyclotomic and $p$-adic directions of Perrin-Riou's pencil, and with $v_p(j)=1$ the central values $M_1,\dots,M_m$ determine $q_p$ modulo $p^m$. The paper then proves that $q_p$ is the coordinate of two global classes: Rubin's elliptic-unit class $x^*$ sits in $E(\mathbb{Q})\otimes\mathbb{Q}_p$ at $q_p$ times an explicit vector, and the derivative $D=\zeta Q_v$ of the derived Heegner class over an auxiliary imaginary quadratic field $F$ satisfies the exact identity $q_p = \frac{1}{S_F}\big(2\ell_a(\sigma_a)\zeta(1-\alpha^{-1})^2\big)^2$, with $S_F$ an explicit rational number built from the rank-one twist. Consequently $q_p \in \prod_v c_v(E)\cdot(\mathbb{Q}_p^\times)^2$, so the unit $q_p/A_E$ of the $p$-adic Birch-Swinnerton-Dyer formula is a square in $\mathbb{Z}_p^\times$ whenever $\Sha(E/\mathbb{Q})$ is finite; and under the additional hypothesis that $Q_v$ has non-zero formal logarithm, the Bertolini-Darmon-Prasanna $p$-adic $L$-function vanishes to order exactly two, the conjecture of Agboola and Castella holds up to a $p$-adic unit, its exact analytic form is equivalent to $q_p=q_\infty$ (with heights read sesquilinearly), and the anticyclotomic Iwasawa-Greenberg main conjecture holds integrally.
Load-bearing premise
The unconditional consequences, namely the square class of $q_p$, the Agboola-Castella result, and the integral main conjecture, require an auxiliary imaginary quadratic field $F$ satisfying Hypotheses 5.1 for the given pair $(E,p)$, and the paper states in Remark 6.6 that conditions (H4) and (H6) cannot be guaranteed in general and are checked case by case, so if no such $F$ exists those theorems are vacuous.
Editorial extensions
If this is right
- The identity makes $q_p$ computable to any $p$-adic precision from critical Hecke $L$-values and $p$-adic heights with cost of order $\sqrt{N_E}$, reaching conductors far beyond modular symbols; the paper reports 5941 tested pairs, with $q_p\equiv q_\infty \pmod{p^3}$ in all 5520 to which the theorem applies.
- Whenever an auxiliary field $F$ satisfying the hypotheses exists, $q_p$ lies in $\prod_v c_v(E)\cdot(\mathbb{Q}_p^\times)^2$, so the unit $q_p/A_E$ of the $p$-adic Birch-Swinnerton-Dyer formula is a square in $\mathbb{Z}_p^\times$ when $\Sha(E/\mathbb{Q})$ is finite, the first information on that unit beyond its valuation.
- If the derived Heegner class has non-zero formal logarithm, the Bertolini-Darmon-Prasanna $p$-adic $L$-function vanishes to order exactly two and the conjecture of Agboola and Castella holds up to a $p$-adic unit; its exact analytic form, with anticyclotomic heights read sesquilinearly, holds if and only if $q_\infty\in\mathbb{Q}$ and $q_p=q_\infty$.
- Together with the base-change main conjecture after inverting $p$, the leading-coefficient identity upgrades to the integral anticyclotomic Iwasawa-Greenberg main conjecture $\operatorname{char}(\mathcal{X}^{\mathrm{Gr}}) = (L_p^{\mathrm{BDP}}(f))$.
- The numerical evidence includes 226 curves with analytic $\Sha$ of order 4 and 20 with order 9; because 4 and 9 are prime to $p$, these orders are invisible to characteristic-ideal statements and only the exact formula predicts them.
Reading between the lines
- Editorial extension: the same 'constant of the quadratic term' template suggests a test in the remaining class-number-one imaginary quadratic fields, where the paper notes that the simplifying identity $s_2=0$ fails and a reduced theta function replaces the sigma function; whether an analogue of Theorem 3.5 survives there would separate the role of the height pairing from the special geometry of $
- Editorial extension: the square-class theorem acts as a cheap sieve, since a pair $(E,p)$ whose Tamagawa product is not a square in $\mathbb{Q}_p$ but whose computed $q_p$ still satisfies $q_p\equiv q_\infty\pmod{p}$ would expose a tension between the conjecture and the arithmetic BSD factors, and this can be checked before any expensive central-value computation.
- Editorial extension: the sign between the literal and sesquilinear readings of the Agboola-Castella products is empirically decidable without knowing $q_p$; for any new pair $(E,F,p)$, computing $\Xi^2$ from Heegner points alone and comparing with $\pm q_\infty s_F$ would settle the convention, and the two treated examples agree with the sesquilinear choice.
- Editorial extension: the author's observation that weight $k=2$ is the only computed weight where $L(f_k/F,s)$ vanishes to order three suggests a deformation probe of the rationality of $q_\infty$; if a computation along the Hida family found another weight with order-three vanishing, the coordinate $\zeta$ would be predicted rational there as well, linking the derived-class identity to a one-para
Original abstract
Let $E\colon y^2=x^3-Dx$ have rank two and let $p\equiv1\pmod4$ be a prime of good reduction. We study the conjecture that the $p$-adic quotient $q_p$, the leading coefficient of the cyclotomic $p$-adic $L$-function divided by the Euler factor and the $p$-adic regulator, equals the complex quotient $q_\infty=L''(E,1)/(2\Omega_E\mathrm{Reg}_\infty)$, and that the latter is rational. In rank one the analogous statement is a theorem of Perrin-Riou and Rubin, because a Heegner point computes both sides; in rank two there is no such point. We show that $q_p$ is the constant of the quadratic term of the two-variable Katz measure; this is how the conjecture arises, and it computes $q_p$ modulo any power of $p$ from critical Hecke $L$-values. We then show that $q_p$ is the coordinate of two global classes: Rubin's rank-two elliptic-unit class, whose position in $E(\mathbb{Q})\otimes\mathbb{Q}_p$ is $q_p$ times an explicit vector, and the derived Heegner class over an auxiliary imaginary quadratic field $F$, for which we prove an exact identity. As consequences, whenever such a field $F$ satisfying our hypotheses exists, $q_p$ is a square in $\mathbb{Q}_p$ times the product of the Tamagawa numbers of $E$, so that the unit $q_p/A_E$ of the $p$-adic Birch and Swinnerton-Dyer formula is a square when Sha is finite; if moreover the derived Heegner class has non-zero formal logarithm, the conjecture of Agboola and Castella for $E/F$ holds up to a $p$-adic unit, and with a theorem of Burungale, Castella and Skinner the anticyclotomic main conjecture holds integrally; and our conjecture is equivalent to the analytic form of theirs, with the products of anticyclotomic heights read sesquilinearly, which differs by a sign from its literal formulation. The conjecture is verified modulo $p^3$ for 5520 pairs $(E,p)$ with conductors up to $3.9\cdot10^{12}$, including curves with analytic Sha of order 4 and 9.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Let $E\colon y^2=x^3-Dx$ have rank two and let $p\equiv1\pmod4$ be a prime of good reduction. We study the conjecture that the $p$-adic quotient $q_p$, the leading coefficient of the cyclotomic $p$-adic $L$-function divided by the Euler factor and the $p$-adic regulator, equals the complex quotient $q_\infty=L''(E,1)/(2\Omega_E\mathrm{Reg}_\infty)$, and that the latter is rational. In rank one the analogous statement is a theorem of Perrin-Riou and Rubin, because a Heegner point computes both sides; in rank two there is no such point. We show that $q_p$ is the constant of the quadratic term of the two-variable Katz measure; this is how the conjecture arises, and it computes $q_p$ modulo any power of $p$ from critical Hecke $L$-values. We then show that $q_p$ is the coordinate of two global classes: Rubin's rank-two elliptic-unit class, whose position in $E(\mathbb{Q})\otimes\mathbb{Q}_p$ is $q_p$ times an explicit vector, and the derived Heegner class over an auxiliary imaginary quadratic field $F$, for which we prove an exact identity. As consequences, whenever such a field $F$ satisfying our hypotheses exists, $q_p$ is a square in $\mathbb{Q}_p$ times the product of the Tamagawa numbers of $E$, so that the unit $q_p/A_E$ of the $p$-adic Birch and Swinnerton-Dyer formula is a square when Sha is finite; if moreover the derived Heegner class has non-zero formal logarithm, the conjecture of Agboola and Castella for $E/F$ holds up to a $p$-adic unit, and with a theorem of Burungale, Castella and Skinner the anticyclotomic main conjecture holds integrally; and our conjecture is equivalent to the analytic form of theirs, with the products of anticyclotomic heights read sesquilinearly, which differs by a sign from its literal formulation. The conjecture is verified modulo $p^3$ for 5520 pairs $(E,p)$ with conductors up to $3.9\cdot10^{12}$, including curves with analytic Sha of order 4 and 9.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $E\colon y^2=x^3-Dx$ have rank two and let $p\equiv1\pmod4$ be a prime of good reduction. | background 1.00 | 0.08 | 0.02 | asserted 0.76 |
| s1 | We study the conjecture that the $p$-adic quotient $q_p$, the leading coefficient of the cyclotomic $p$-adic $L$-function divided by the Euler factor and the $p$-adic regulator, equals the complex quotient $q_\infty=L''(E,1)/(2\Omega_E\mathrm{Reg}_\infty)$, and that the latter is rational. | contribution 0.67 | 0.15 | 0.19 | speculative 0.89 |
| s2 | In rank one the analogous statement is a theorem of Perrin-Riou and Rubin, because a Heegner point computes both sides; in rank two there is no such point. | gap 0.54 | 0.05 | 0.04 | asserted 1.00 |
| s3 | We show that $q_p$ is the constant of the quadratic term of the two-variable Katz measure; this is how the conjecture arises, and it computes $q_p$ modulo any power of $p$ from critical Hecke $L$-values. | result 0.62 | 0.93 | 0.12 | asserted 1.00 |
| s4 | We then show that $q_p$ is the coordinate of two global classes: Rubin's rank-two elliptic-unit class, whose position in $E(\mathbb{Q})\otimes\mathbb{Q}_p$ is $q_p$ times an explicit vector, and the derived Heegner class over an auxiliary imaginary quadratic field $F$, for which we prove an exact identity. | result 0.88 | 0.95 | 0.34 | asserted 1.00 |
| s5 | As consequences, whenever such a field $F$ satisfying our hypotheses exists, $q_p$ is a square in $\mathbb{Q}_p$ times the product of the Tamagawa numbers of $E$, so that the unit $q_p/A_E$ of the $p$-adic Birch and Swinnerton-Dyer formula is a square when Sha is finite; if moreover the derived Heegner class has non-zero formal logarithm, the conjecture of Agboola and Castella for $E/F$ holds up to a $p$-adic unit, and with a theorem of Burungale, Castella and Skinner the anticyclotomic main conjecture holds integrally; and our conjecture is equivalent to the analytic form of theirs, with the products of anticyclotomic heights read sesquilinearly, which differs by a sign from its literal formulation. | interpretation 0.76 | 0.84 | 0.43 | asserted 0.97 |
| s6 | The conjecture is verified modulo $p^3$ for 5520 pairs $(E,p)$ with conductors up to $3.9\cdot10^{12}$, including curves with analytic Sha of order 4 and 9. | result 0.99 | 0.91 | 0.13 | asserted 1.00 |
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