REVIEW 4 major objections 5 minor 19 references
Infinitely many elliptic curves over $\mathbb{Q}(i)$ with rank 2 and $j$-invariant 1728
T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read Infinitely many elliptic curves over the Gaussian rationals with j-invariant 1728 have rank exactly 2 and are not base changes from the rationals.
desk verdict The rank-2 construction is genuinely new and the descent looks sound, but the infinitude claim rests on an unproved strengthening of Tao's theorem (dilation in Z rather than Z[i]) that the paper merely asserts. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The argument rests on three mechanisms. The curve is a quartic twist $E_{\alpha}:y^{2}=x^{3}+\alpha x$ with $\alpha=-(\beta^{4}+4k^{4})^{2}$, equipped with a degree-2 isogeny $\varphi:E_{\alpha}\to E_{-4\alpha}$; the $\varphi$-Selmer group, the Galois-cohomology group that bounds the image of $\mathbb{Q}(i)$-points under descent, gives the upper bound on rank. Under the congruence hypotheses, the four primes $p_{j}=\beta+i^{j}k(1+i)$ have prescribed quadratic-residue symbols via quartic reciprocity, so the Selmer matrix $L$ is one of two explicit $4\times4$ matrices over $\mathbb{F}_{2}$; solving $L\vec{1}(d)=0$ or $L\vec{1}(d)=(n_{p_{j}})_{j}$ shows the $\varphi$-Selmer group has dimension $2$, hence rank at most $2$. A non-torsion point, together with evenness of the rank coming from complex multiplication by $\mathbb{Z}[i]$, gives rank at least $2$. Infinitude is carried by the constellation theorem for Gaussian primes: the class of primes congruent to $-1-6i$ modulo $16$ has natural density $1/128$, hence positive upper relative Banach density, so configurations $\beta+k\gamma_{j}$ with $\gamma_{j}=i^{j}(1+i)$ occur for infinitely many pairs $(\beta,k)\in\mathbb{Z}[i]\times\mathbb{Z}$.
What would settle it
Take any pair $(\beta,k)$ whose four Gaussian integers are Gaussian primes congruent to $-1-6i$ modulo $16$, then compute the $\varphi$-Selmer group and search for rational points; if the $\varphi$-Selmer dimension exceeds $2$, or if three independent non-torsion $\mathbb{Q}(i)$-points are found, the rank is forced above $2$ and the theorem's conclusion fails.
Extended reading notes
Core claim
Let $\beta\in\mathbb{Z}[i]$ and $k\in\mathbb{Z}$ be such that the four Gaussian integers $\beta+k(1+i)$, $\beta+ki(1+i)$, $\beta-k(1+i)$, $\beta-ki(1+i)$ are all Gaussian primes congruent to $-1-6i$ modulo $16$. Then the elliptic curve $E:y^{2}=x^{3}-(\beta^{4}+4k^{4})^{2}x$ over $\mathbb{Q}(i)$ satisfies $E(\mathbb{Q}(i))\cong\mathbb{Z}^{2}\oplus(\mathbb{Z}/2\mathbb{Z})^{2}$ and is genuinely defined over $\mathbb{Q}(i)$: its defining coefficient is not a rational integer, so the curve is not the base change of a curve over $\mathbb{Q}$. The rank is bounded above by computing that the $\varphi$-Selmer group associated to the $2$-isogeny has dimension $2$, while the explicit point $(4\beta^{2}k^{2},\,2i\beta k(\beta^{4}-4k^{4}))$ is non-torsion and the rank is even because of complex multiplication by $\mathbb{Z}[i]$, forcing rank exactly $2$. The constellation theorem for Gaussian primes supplies infinitely many such pairs $(\beta,k)$, hence infinitely many such curves.
Load-bearing premise
The entire infinitude step rests on an imported strengthened constellation theorem asserting that any Gaussian-prime set of positive upper relative Banach density contains infinitely many configurations $\beta+k\gamma_{j}$ with the dilation $k$ an ordinary integer, and the paper cites existing sources for this strengthening rather than proving it.
Editorial extensions
If this is right
- For every eligible pair $(\beta,k)$, the curve $y^{2}=x^{3}-(\beta^{4}+4k^{4})^{2}x$ has rank exactly $2$ over $\mathbb{Q}(i)$, so the upper and lower rank bounds coincide in this family.
- The construction yields infinitely many curves with $j$-invariant $1728$ that are genuinely defined over $\mathbb{Q}(i)$; none is a base change from $\mathbb{Q}$.
- Every curve in the family has full rational $2$-torsion, $(\mathbb{Z}/2\mathbb{Z})^{2}$, and no $4$-torsion, so the torsion group is exactly the minimal one allowed by the $2$-torsion.
- Because the rank of a curve with complex multiplication by $\mathbb{Z}[i]$ over $\mathbb{Q}(i)$ is even, the descent bound exactly matches the parity lower bound in this family.
Reading between the lines
- The same two-ingredient recipe, a rational $2$-isogeny and a positive-density prime class, should produce rank-2 families over other imaginary quadratic fields, although the Selmer matrix computation would need to be redone for each field.
- A quantitative strengthening of the constellation theorem would give lower bounds on the number of such rank-2 curves with bounded coefficients; the paper only establishes infinitude.
- Other residue classes of Gaussian primes whose Selmer matrix can be analyzed may yield additional infinite families of genuinely defined rank-2 curves, possibly with different torsion groups.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies elliptic curves over Q(i) with j-invariant 1728, written in the quartic twist form E: y^2 = x^3 - (β^4 + 4k^4)^2 x with β ∈ Z[i] and k ∈ Z. The main theorem, Theorem 1.2, asserts that if the four Gaussian integers β + i^j k(1+i) are primes congruent to -1-6i modulo 16, then E is genuinely defined over Q(i) and E(Q(i)) ≅ Z^2 ⊕ (Z/2Z)^2. The proof computes the Q(i)-torsion by division polynomials, exhibits a non-torsion point, and bounds the rank from above using a 2-isogeny descent whose Selmer computation is imported from [KS24]. Theorem 1.1 then claims that infinitely many such curves arise by applying a strengthened form of Tao's Gaussian constellation theorem, stated as Theorem 4.1. The rank and torsion portions of the paper are explicit and mostly checkable, but the infinitude step depends on a nontrivial strengthening of Tao's theorem that is cited rather than proved, and Theorem 1.2 as stated admits a degenerate k=0 case in which the desired conclusion is not established.
Significance. If the proof can be completed, the result would provide a natural infinite family of elliptic curves over Q(i) with j-invariant 1728, rank exactly 2, and no base-change origin; this would be a worthwhile complement to Zywina's rational rank-2 construction and would illustrate the interaction of descent methods with additive combinatorics over number fields. The paper's torsion analysis (Theorem 2.3), the non-torsion point in the proof of Theorem 1.2, and the residue-symbol computation in Lemma 3.4 are concrete and internally coherent, and the Selmer-bound computation is explicit enough to be verified. The main reservations concern the unproved strengthened constellation theorem and the degenerate k=0 case; these are load-bearing for the infinitude theorem and for the correctness of Theorem 1.2 as stated.
major comments (4)
- [Section 4, Theorem 4.1] The statement requires the dilation parameter k to lie in Z, which is strictly stronger than the usual form of Tao's Gaussian constellation theorem, where the dilation is allowed to lie in Z[i]. The proof consists only of a citation to [Tao06, Section 12] and the assertion that this is a special case of [KMM+20, Theorem 1.4]; no derivation is given. If the cited results only produce k ∈ Z[i], then Theorem 4.1 is unsupported and the infinitude step in the proof of Theorem 1.1 has no foundation. The authors should either prove Theorem 4.1 in the manuscript or quote the exact statement from the cited papers that supplies rational dilation, together with enough detail to verify it.
- [Section 4, Theorem 4.1 and proof of Theorem 1.1] As stated, Theorem 4.1 does not exclude k = 0. Since P is infinite, every pair (β, 0) with β ∈ P satisfies the conclusion, so the theorem is then trivially true and does not imply that the rank-2 curves of Theorem 1.2 occur infinitely often. The application in the proof of Theorem 1.1 needs infinitely many pairs with k ≠ 0; the theorem should state this explicitly (as is standard for constellation theorems), and the cited sources should be checked to ensure that they indeed give such pairs.
- [Theorem 1.2] The hypotheses of Theorem 1.2 admit k = 0: if β is any Gaussian prime congruent to -1-6i modulo 16, then the four listed integers are all equal to β and hence are primes. In this case the p_j are not distinct, the proposed point (4β²k², 2iβk(β⁴-4k⁴)) is (0,0), and Theorem 3.3, which requires distinct primary primes, cannot be applied. Moreover, for k = 0 the curve is isomorphic over Q(i) to E_{-1}: y² = x³ - x, which has rank 0 over Q(i), so the conclusion "rank exactly 2" is false as stated. The theorem must require k ≠ 0, equivalently that the four primes are distinct.
- [Proof of Theorem 1.2, genuinely defined argument] The argument that E is not a base change contains two incorrect statements. From p_1² p_2² p_3² p_4² ∈ Z one obtains that the set {p_j} is closed under complex conjugation, so ̅p_1 = p_l for some l, not p_1 ∈ {p_2, p_3, p_4}. Also, with the labeling p_j = β + i^j k(1+i) used in Lemma 3.4, the equality p_1 = p_2 forces k = 0, not Im(β) = 0. The intended conclusion can be recovered by observing that the only nontrivial conjugate pairing forces β to be real, contradicting Im(β) ≡ 2 (mod 8), but the written proof needs to be corrected.
minor comments (5)
- [Title] The full-text title contains a typo: "INFINITEL Y MANY" should read "INFINITELY MANY".
- [Theorem 1.2 and Lemma 3.4] The ordering of the four primes in Theorem 1.2 (k, ik, -k, -ik) differs from the p_j ordering in Lemma 3.4, where p_j = β + i^j k(1+i) for j = 1,...,4; define the p_j explicitly in the statement of Theorem 1.2 to avoid ambiguity.
- [Lemma 3.4] The notation (1+i/p_4)_2^2 is easy to misread; write the square of the quadratic residue symbol explicitly, for example (1+i)/p_4_2 squared, to avoid confusion with an exponent on the denominator.
- [Proof of Theorem 3.3] The paragraph importing Theorem 3.3 from [KS24] is very terse; for readers not familiar with that preprint, a fuller explanation of how the two cases of S′ arise from the local solubility conditions would improve readability and verifiability.
- [Proof of Theorem 1.1] In the density computation, it would be helpful to state explicitly that the class group quotient by the units {±1, ±i} has size 32, so that the Dirichlet density 1/32 and the subsequent factor 1/4 for primary associates are transparent.
Circularity Check
No significant circularity: the rank computation and infinitude proof are self-contained relative to stated external theorems.
full rationale
The paper's central derivation does not reduce to its inputs. Theorem 1.2 gives an explicit non-torsion point (4β^2k^2, 2iβk(β^4−4k^4)) on E, establishing a lower bound of 2 because CM forces even rank; this is independent of the Selmer upper bound. The upper bound is obtained by specializing the general φ-Selmer computation of [KS24] to α=−p1²p2²p3²p4²; [KS24] is parameter-free and does not assume the rank-2 conclusion, so this reliance is ordinary theorem use. Lemma 3.4 computes the residue-symbol matrix from quartic reciprocity, and the resulting bound rank ≤2 is then combined with the lower bound; no fitted parameter is relabeled as a prediction. The infinitude step imports Tao's Gaussian-prime constellation theorem and density results from Chebotarev; the paper explicitly flags that its Theorem 4.1 is stronger than the published statement, which is a verification/correctness concern rather than circularity. The only self-citation is [KS24], a prior preprint by one of the authors, but it supplies a general algorithm, not the target theorem, so it does not make the argument circular.
Assumptions & free parameters
assumptions (6)
- domain assumption Torsion classification for elliptic curves over Q(i) (Najman, Theorem 2.1).
- domain assumption Theorem 3.3, the phi-Selmer group algorithm for Q(i) from [KS24, Theorem 5.6] with alpha of square type.
- domain assumption Theorem 4.1, the strengthened Gaussian-prime constellation theorem with rational integer dilation k and positive relative Banach density subsets.
- standard math Quartic reciprocity and primary factorization for Gaussian integers (Lemma 3.2, [Lem13, Prop 6.8]).
- standard math Generalized Dirichlet density and Chebotarev density for ray classes in Q(i).
- standard math Z[i] is a UFD and the elliptic curve facts from Silverman (2-isogeny descent exact sequence, quartic twists).
Cite this review
Pith. "Pith review of Infinitely many elliptic curves over $\mathbb{Q}(i)$ with rank 2 and $j$-invariant 1728." pith.science (2026). https://pith.science/paper/7YMDW7G3
@misc{pith2026250617605,
author = {Pith},
title = {Pith review of: Infinitely many elliptic curves over $\mathbbQ(i)$ with rank 2 and $j$-invariant 1728},
year = {2026},
howpublished = {\url{https://pith.science/paper/7YMDW7G3}},
note = {Machine review of arXiv:2506.17605}
}
abstract
We prove that there exist infinitely many elliptic curves over $\mathbb{Q}(i)$ with $j$-invariant $1728$ and rank exactly $2$ which are not obtained by base change from $\mathbb{Q}$. The rank of each such curve is determined via 2-isogeny descent, and the existence of infinitely many such curves follows from Tao's constellation theorem for Gaussian primes.
Reference graph
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Reviewed August 15, 2026 · model on record in the stance chip above.
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