A normalization issue in a transmitted form of Bloch's diamond formula: anti-invariant subgroup versus regulator quotient

Huimin Zheng

REVIEW

A normalization issue in a transmitted form of Bloch's diamond formula: anti-invariant subgroup versus regulator quotient

Huimin Zheng

Number Theory · Submitted 2 Nov 2026

Pith's verdict: A clean, honest note correcting a factor-2 ambiguity in the transmitted Bloch–diamond formula; the derivation is elementary (one homology-index computation from Brunault's published result), the numerical evidence is honest, and the limitations are stated.

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arXiv ID2611.05420 (version 2)

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CitationHuimin Zheng. "A normalization issue in a transmitted form of Bloch's diamond formula: anti-invariant subgroup versus regulator quotient." arXiv:2611.05420 (2026). https://pith.science/paper/OFM3P5PA

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Editor's note

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Huimin Zheng has written a short, focused note that does exactly what it says: it identifies a normalization mismatch between the anti-invariant subgroup and the regulator quotient in how Bloch's diamond formula has been transmitted in the Mahler-measure literature, and it states the corrected factor ι(E) (1 or 2). The core observation is elementary—a two-line index computation from Brunault's published intersection-product formula—and the paper is honest about that. It gives credit where it is due: the factor was already implicit in Brunault's own writing (E⁺·γ = −2 on curves with Δ<0), and no claimed theorem is refuted; the paper just points out that the literal sentence 'γ a generator of H₁(E,Z)⁻' in the transmitted form is off by a factor of 2 on curves with one real component.

The strengths are brevity, clarity, and candor. The numerical check on 20.a4 is decent (13 digits, not interval-certified but honestly described as high-precision confirmation), and the two proved examples (11.a3 via the author's own submitted work, and the consistency check on Touafek's Theorem 2) show the paper understands its own limitations. The paper does not overclaim—it explicitly says the literature audit is incomplete and that Touafek's final D_E-relation is not contradicted.

Soft spots: the central claim is essentially a footnote to Brunault, and the novelty is modest—the correction is known to experts who have worked through the normalization. The paper's own main example (11.a3) depends on the author's submitted arXiv:2609.26119; the paper acknowledges this and says no circularity arises because Theorem 4.1 is proved from Brunault alone. That's fine, but it means one of the two documentary cases is not fully self-contained. The analysis of Touafek's Theorem 2 identifies a "two-part reinterpretation" but leaves the derivation of the 1/2 factor in Touafek's formula (5) as an open question—the paper is honest about this, but it means that part is diagnostic, not definitive.

Overall the paper does what it sets out to do: it clarifies an ambiguity in the transmitted literature. It is not ground-shaking, but it is correct and useful for people who work with Bloch's formula on curves with Δ<0. I would send it to referee for a specialized venue; it deserves a serious editor to decide if it rises to a full paper or is better as a short note.

Referee report

Significance

The correction is important because the misstated form of Bloch's diamond formula appears in several papers in the Mahler-measure literature (e.g., Lalin-Ramamonjisoa). The paper shows that no published identity is actually invalidated---all checked examples with Delta>0 are fine, and those with Delta<0 that were stated as D_E-value identities do not rely on the erroneous wording---but the record must be set straight for future work. The derivation is short, rigorous, and relies on a single published external result (Brunault, 2016); the numerical verification is high-precision and the proved example on 11.a3 (using the regulator anchor of the companion paper) is independent of the main result. The paper thus clears up a persistent ambiguity in the literature at very low mathematical cost.

Minor objections (5)

[Lemma 3.1] The description of the homology basis for Delta<0 is correct, but a brief justification of why c_*b = a-b (e.g., via the shape of the period lattice) would help readers less familiar with real elliptic curves. The current phrasing is sufficient, but an explicit coordinate description of the generators would improve clarity.
[Lemma 3.3] The proof that the pairing gamma·delta_tilde is well-defined could be spelled out: because c_*gamma=gamma and c_*delta_plus=delta_plus, the intersection number changes sign under the anti-holomorphic involution, forcing gamma·delta_plus=0. The present one-line argument is essentially this, but a sentence making the sign reasoning explicit would remove any possible doubt.
[Remark 5.2] The finite certificates for the split-integral argument are sketched but not fully verified within this paper; the text already notes they are finite replacements independent of period certification. It would be helpful to add a sentence confirming that these arguments are not relied upon for the main theorem, which is proved from Brunault's Proposition 19 alone.
[Section 6] The numerical verification is honestly described as high-precision but not interval-certified, and the residual is consistent with the expected error. To avoid any misinterpretation, consider adding a brief remark that the extrapolation step assumes the error series is asymptotic; the observed residual supports this but does not prove it.
[Throughout] The notation iota(E) is defined only in the abstract and in (4); it would be convenient to restate its definition where it first appears in the body (Section 3 or Section 4), since the abstract is not part of the formal proof.

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The paper

The paper claims

Bloch's diamond formula carries a hidden factor of 2 for elliptic curves with one real component.

The reading

Bloch's formula relating the regulator of a $K_2$ symbol on an elliptic curve to the elliptic dilogarithm of a 'diamond' product is widely cited with the integral taken over a generator of the anti-invariant homology subgroup. This paper shows that this transmitted form is incorrect when the real locus of the curve has one connected component, i.e., when the discriminant is negative. The correct identity requires a factor of $2$ if the same subgroup generator is used; the natural normalization uses the regulator quotient, where the factor is $1$. The discrepancy arises because the anti-invariant subgroup has index $2$ in the regulator quotient when $\Delta<0$. The paper proves the corrected general statement, documents it on two curves with $\Delta<0$ using both proved and high-precision numerical values, and identifies a published identity that mixes two independent normalizations.

What carries the argument

The regulator quotient $Q(E)=H_1(E,\mathbb{Z})/H_1(E,\mathbb{Z})^+$, a rank-$1$ free abelian group on which the regulator integral is naturally defined. Brunault's intersection-product formula (Proposition 19) expresses the regulator integral as $-(E^+\cdot\gamma)$ times the diamond value, where $E^+$ generates $H_1(E,\mathbb{Z})^+$. By Lemma 3.3, a generator of $Q(E)$ lifts to a cycle $\gamma$ with $E^+\cdot\gamma=\pm1$, giving the factor-$1$ identity. The index computation (Lemma 3.2) shows that the anti-invariant subgroup maps into $Q(E)$ with index $\iota(E)$ as described, yielding the factor $2$ for $\Delta<0$ curves.

Extended reading notes

Core claim

The paper establishes that Bloch's diamond formula, when stated for a generator of the anti-invariant subgroup $H_1(E,\mathbb{Z})^-$, holds with an extra factor $\iota(E)\in\{1,2\}$ that equals $1$ when the real curve has two components ($\Delta>0$) and $2$ when it has one component ($\Delta<0$). The natural domain of the regulator integral is the regulator quotient $Q(E)=H_1(E,\mathbb{Z})/H_1(E,\mathbb{Z})^+$, where the factor is $1$ for a generator (Theorem 4.1). The subgroup version follows by an elementary index computation: for $\Delta<0$, the class of the anti-invariant generator equals $\pm 2$ times the quotient generator in $Q(E)$ (Lemma 3.2). Consequently, the transmitted form in Lalín–Ramamonjisoa and elsewhere is correct only when $\Delta>0$; for $\Delta<0$ it is off by a factor of $2$. The paper also shows that Touafek's Theorem 2 mixes two independent normalizations and becomes consistent only after a two-part reinterpretation.

Load-bearing premise

The proof treats Brunault's intersection-product formula (Proposition 19) as a black box, relying on its intrinsic normalization of the generator $E^+$ and orientation conventions; if that formula had a normalization different from what the paper assumes, the derived factor could shift.

Editorial extensions

If this is right

Any regulator identity cited with the subgroup wording and factor $1$ should be rechecked for $\Delta<0$ curves; the factor $2$ must be inserted.
The main theorem of Lalín–Ramamonjisoa (conductor $17$, $\Delta>0$) is unaffected because subgroup and quotient coincide.
Bertin's proved exotic relations among $D_E$ values involve diamond values alone, not regulator integrals, so they are unaffected.
Touafek's final $D_E$-relation is not contradicted by the lattice correction, though its printed derivation retains a gap now identified.
Future work on $\Delta<0$ curves should use the quotient generator or apply Corollary 4.2 to avoid sign or factor errors.

Reading between the lines

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

The factor $2$ for $\Delta<0$ curves may resolve apparent discrepancies in numerical verifications of Bloch's formula in the Mahler-measure literature that have used the subgroup wording without checking the real topology.
The finding suggests that earlier works (e.g., Deninger, Rodríguez Villegas) may have implicitly assumed a two-component real locus; a systematic audit of each cited curve's discriminant could clarify the history.
The open question about Touafek's formula (5) under the quotient reading invites recomputing the $2$-isogeny action on the quotient lattice, which could clarify other isogeny-based regulator identities and resolve the derivation gap.

Original abstract

Bloch's regulator formula for an elliptic curve is often quoted with a generator of the anti-invariant subgroup $H_1(E,\mathbb{Z})^-$ and coefficient $1$ in front of the elliptic dilogarithm of the diamond product. We compare this wording with Brunault's published intersection-product formula. The natural map from the anti-invariant subgroup to the regulator quotient $H_1(E,\mathbb{Z})/H_1(E,\mathbb{Z})^+$ has index $1$ when $E(\mathbb{R})$ has two components and index $2$ when it has one. Thus, up to orientation, the coefficient is $1$ for a quotient generator and $\iota(E)\in\{1,2\}$ for a subgroup generator. This is an elementary consequence of the published formula, not a new regulator theorem. We illustrate the distinction on 11.a3 using an externally proved regulator anchor and the Brunault--Bertin special values, and on 20.a4 by high-precision numerical integration. We also correct our earlier quotation of Touafek: his printed identity is $\pi r=D_E((f)\diamond(g))$, with $r=(2\pi)^{-1}\int_{\gamma^-}\eta$. It is compatible with the subgroup normalization when $\Delta<0$; when $\Delta>0$ the corresponding coefficient is $1/2$, up to orientation. We do not reassess his isogeny theorem or his final exotic relation here. No new proof of Boyd's conductor-11 split-integral identity is included.

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.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-05: 1 request, 6,516 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.96 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.92 · compares with prior results 0.96 · mentions code or data availability 0.05.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Bloch's identity $\int_\gamma\eta(f,g)=D_E((f)\diamond(g))$, relating the regulator of a $K_2$-symbol on an elliptic curve $E/\mathbb{Q}$ to the elliptic dilogarithm of the diamond product of the divisors, is transmitted in the Mahler-measure literature with $\gamma$ generating the anti-invariant subgroup $H_1(E,\mathbb{Z})^-$ and factor $1$ (e.g. Lalin-Ramamonjisoa, Thm. 6). We observe that, when $E(\mathbb{R})$ has one connected component ($\Delta<0$), this reading is incompatible with the published intersection-product normalization (Brunault, Proposition 19): the anti-invariant subgroup then has index $2$ in the regulator quotient $H_1(E,\mathbb{Z})/H_1(E,\mathbb{Z})^+$. The correct statement -- factor $1$ for the quotient generator, hence factor $\iota(E)\in\{1,2\}$ for the subgroup generator, with $\iota(E)=1\iff\Delta>0$ -- follows from Brunault's formula by an elementary index computation. We document the phenomenon twice: on 11.a3 by proved values (a Siegel-unit anchor from [Zheng, arXiv:2609.26119]; the diamond side by Brunault and Bertin), and on 20.a4 by high-precision computation of both integrals against the $q$-series for $D_E$ (subgroup/quotient ratio $2$ to $13$ digits). We also show that Touafek's Theorem 2 (involving a $\Delta<0$ curve) mixes two independent normalizations (subgroup versus quotient, and a $(2\pi)^{-1}$ prefactor versus the bare diamond value), and we identify the two-part reinterpretation making its printed formulas mutually compatible. In the cases we have checked, no Mahler-measure or $D_E$-identity is affected: the main theorem of Lalin-Ramamonjisoa (conductor $17$, $\Delta>0$) and Bertin's proved exotic relations stand untouched, and Touafek's final $D_E$-relation is not contradicted and is numerically supported, though its printed derivation retains the gap described in the text. A systematic audit is beyond the scope of this note.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Bloch's identity $\int_\gamma\eta(f,g)=D_E((f)\diamond(g))$, relating the regulator of a $K_2$-symbol on an elliptic curve $E/\mathbb{Q}$ to the elliptic dilogarithm of the diamond product of the divisors, is transmitted in the Mahler-measure literature with $\gamma$ generating the anti-invariant subgroup $H_1(E,\mathbb{Z})^-$ and factor $1$ (e.g. Lalin-Ramamonjisoa, Thm. 6).	background 1.00	0.04	0.05	asserted 1.00
s1	We observe that, when $E(\mathbb{R})$ has one connected component ($\Delta<0$), this reading is incompatible with the published intersection-product normalization (Brunault, Proposition 19): the anti-invariant subgroup then has index $2$ in the regulator quotient $H_1(E,\mathbb{Z})/H_1(E,\mathbb{Z})^+$.	result 0.86	0.74	0.23	asserted 1.00
s2	The correct statement -- factor $1$ for the quotient generator, hence factor $\iota(E)\in\{1,2\}$ for the subgroup generator, with $\iota(E)=1\iff\Delta>0$ -- follows from Brunault's formula by an elementary index computation.	result 0.86	0.77	0.78	asserted 1.00
s3	We document the phenomenon twice: on 11.a3 by proved values (a Siegel-unit anchor from [Zheng, arXiv:2609.26119]; the diamond side by Brunault and Bertin), and on 20.a4 by high-precision computation of both integrals against the $q$-series for $D_E$ (subgroup/quotient ratio $2$ to $13$ digits).	contribution 0.56	0.92	0.11	asserted 1.00
s4	We also show that Touafek's Theorem 2 (involving a $\Delta<0$ curve) mixes two independent normalizations (subgroup versus quotient, and a $(2\pi)^{-1}$ prefactor versus the bare diamond value), and we identify the two-part reinterpretation making its printed formulas mutually compatible.	result 0.48	0.90	0.08	asserted 1.00
s5	In the cases we have checked, no Mahler-measure or $D_E$-identity is affected: the main theorem of Lalin-Ramamonjisoa (conductor $17$, $\Delta>0$) and Bertin's proved exotic relations stand untouched, and Touafek's final $D_E$-relation is not contradicted and is numerically supported, though its printed derivation retains the gap described in the text.	interpretation 0.48	0.70	0.21	asserted 0.87
s6	A systematic audit is beyond the scope of this note.	limitation 1.00	0.26	0.02	asserted 0.92

What kind of paper the classifier reports this is: position or commentary 0.61 · evidence from proof or derivation 0.89.

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