REVIEW 3 major objections 4 minor 21 references
Elliptic Curves, Riordan arrays and Lattice Paths
T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read For every curve of the form $y^2-axy-y=x^3-bx^2-cx$, the Hankel transform of the lattice-path enumerating sequence is, up to initial terms, the elliptic divisibility sequence of the curve.
desk verdict A promising general construction linking elliptic curves to lattice paths via Riordan arrays, but the central Hankel/EDS identification is asserted without proof and the general reversion formula has a sign error. 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 engine is the A-matrix characterization of Riordan arrays. From a recurrence of the form $t_{n,k}=t_{n-1,k-1}+\alpha t_{n-1,k}+\beta t_{n-2,k}+\gamma t_{n-2,k-1}+\delta t_{n-2,k+1}$, the theory gives $u/x=1+\gamma x+\alpha u+\beta ux+\delta u^2x$, whose solution yields $g(x)=u/x$ in the Catalan form above. Matching $\alpha,\beta,\gamma,\delta$ to the curve's parameters produces the step sets, and the Hankel transform of $g$ is then the Somos 4 sequence $(\delta^2,\ \delta^2(\alpha\gamma-\beta+\gamma^2))$, which here becomes $(1,-ac+b+c^2)$. On the curve side the classical elliptic divisibility sequence is also Somos 4; the bridge is the claim that the two initial data agree. The generating function $g$ also has a continued fraction whose coefficients are the coordinates of the multiples $nP$ of $P=(0,0)$, linking the group law to the combinatorics.
What would settle it
Take the paper's Example 2 curve $y^2+2xy-y=x^3+5x^2-x$ and compute the Hankel transform of the sequence $1,-1,3,2,17,51,185,664,2333,8360,29717,\dots$ beyond the six displayed terms, then compare with the elliptic divisibility sequence at $(0,0)$, computed from the division polynomials. If any term beyond the displayed initial segment fails to match after the stated shift, the essentially-coincides claim is false; matching to 15 or 20 terms would strengthen but not prove it.
Extended reading notes
Core claim
The central claim, stated the way a sympathetic reader would state it: for parameters $a,b,c$, the reversion process applied to the curve equation produces $$g(x)=\frac{1+(a-2c+1)x}{1-x(2(c-1)-a)-$x^{2}$(a(c-1)-b-(c-1)^2)} C\left(\frac{$x^{3}$(1-x(a-2c+1))}{(1-x(2(c-1)-a)-$x^{2}$(a(c-1)-b-(c-1)^2))^2}\right),$$ where $C$ is the Catalan generating function. The expansion $u_n$ of $g(x)$ is the return-count sequence for a family of signed lattice paths with step set $\{(1,1), (2(c-1)-a)*(1,0), (a(c-1)-b-(c-1)^2)*(2,0), (a-2c+1)*(2,1), (2,-1)\}$. Its Hankel transform begins $1,\ ac-b-c^2,\ a^2c-a(b+3c^2)+2bc+2c^3-1,\dots$, and the paper claims this is a $(1,-ac+b+c^2)$ Somos 4 sequence that essentially coincides with the elliptic divisibility sequence of the curve at $(0,0)$. The worked examples make the coincidence explicit: for $y^2+xy-y=x^3+2x^2+x$ the Hankel transform begins $1,2,1,-7,-16,-57,-113,\dots$ while the elliptic divisibility sequence begins $0,1,1,2,1,-7,-16,-57,-113,\dots$; for $y^2+2xy-y=x^3+5x^2-x$ the Hankel transform begins $1,2,-9,-17,-196,593,\dots$ and the divisibility sequence begins $1,1,2,-9,-17,-196,593,\dots$.
Load-bearing premise
The load-bearing premise is that the Hankel transform sequence and the elliptic divisibility sequence are essentially the same: the paper shows the first few terms match, but it does not specify the exact shift or prove that both sequences satisfy the same recurrence from identical starting data.
Editorial extensions
If this is right
- Every curve of the stated form yields an explicit signed lattice-path model, with one step set read from $g(x)$ and another from its binomial transform $\gamma(x)$.
- Because binomial transforms preserve Hankel transforms, the entire family of sequences obtained from $g(x)$ by binomial transforms shares the same Somos 4 Hankel transform, essentially the elliptic divisibility sequence.
- The continued fraction form of $g(x)$ is built from the multiples of $P=(0,0)$, so the group law of the curve is encoded in the path-count generating function.
- When $ac-b-c^2=0$ (for instance $(a,b,c)=(r+1,r,r)$ or $b=0$, $a=c$), the associated Riordan array is a pseudo-involution, giving involutions in the Riordan group from elliptic curves.
- If the identification with elliptic divisibility sequences is made precise, the Hankel transform becomes a combinatorial route to the arithmetic of these curves.
Reading between the lines
- The paper leaves open why the reversion step works; a natural extension is to test the same recipe on general Weierstrass equations and rational base points other than $(0,0)$.
- If the essential coincidence turns out to be an exact equality after a fixed shift, it would give a lattice-path realization of elliptic divisibility sequences and could connect the Laurent phenomenon of Somos 4 sequences with the algebra of walk enumeration.
- Because the paper notes the construction depends on the $j$-invariant equivalence class rather than the particular equation, curves with the same $j$-invariant should yield Hankel-equivalent path models; checking this for several $j$-invariants would be a sharp test.
- The signed step sets suggest an interpretation of the path counts as a determinant or a signed weighting; a bijective reading of the signs could turn the algebraic construction into a combinatorial one.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper associates to each elliptic curve of the form y^2 - a x y - y = x^3 - b x^2 - c x, with the point P=(0,0), a sequence u_n defined through a Riordan array (g(x), x g(x)), whose step set is read off from the curve's parameters. The Hankel transform of u_n is claimed to be a (1, -ac+b+c^2) Somos-4 sequence that 'essentially coincides' with the elliptic divisibility sequence of the curve. Two worked examples are given, together with a general formula for g(x), the associated Riordan array, and an appendix on the A-matrix characterization of Riordan arrays from which the Somos-4 parameters are obtained.
Significance. If the claimed identification were proved, the paper would provide a new combinatorial model for elliptic divisibility sequences: return counts of signed lattice paths whose Hankel transform reproduces the EDS up to an explicit normalization. The explicit formulas are checkable, and the examples give concrete evidence for the plausibility of the connection. The weakness is that the central identification is not established; the paper currently establishes a construction and a family of examples, not a theorem. The A-matrix appendix is a useful compendium of the relevant general facts, and the paper is clearly written for a combinatorial audience.
major comments (3)
- [§3] The displayed reversion expression 2x/(1 + x(a - 2c + 2) + sqrt(1 + 2x(a - 2c) + x^2(a^2 - 4b) + 4x^3)) has the wrong sign in the linear coefficient: direct substitution into the quadratic solution gives 1 + x(2c - 2 - a) in place of 1 + x(a - 2c + 2), and Example 1 (where a = -1, c = -1) confirms the coefficient should be -3, not +3. The same sign error propagates into the displayed general formula for g(x), whose Catalan argument is written with 1 - x(a - 2c + 1) whereas the u-equation given later in the same section and the appendix formula require 1 + x(a - 2c + 1). As written, the general g(x) is not the generating function of the sequence satisfying the stated recurrence, so the construction in the general case is internally inconsistent and must be corrected.
- [§3] The sentence 'This is a (1, -ac+b+c^2) Somos 4 sequence, which essentially coincides with the elliptic divisibility sequence of the elliptic curve' is the load-bearing step of the paper, but it is asserted without proof. The term 'essentially coincides' is never defined: no index shift, sign convention, or normalization of the elliptic divisibility sequence W_n is specified, and the examples check only the first few terms. Since a Somos-4 sequence is determined by four initial values, the three displayed Hankel terms (1, ac-b-c^2, a^2c-a(b+3c^2)+2bc+2c^3-1) are insufficient; the fourth term h_3 is never computed in closed form. A proof would need to identify the Hankel transform with a shifted or normalized EDS by showing equality of the initial data and of the Somos-4 recurrence for all n, or at least state the precise relation as a conjecture. As it stands, this key claim is unproven.
- [§2] The extension of the claimed coincidence to all binomial transforms g_r relies on the assertion that each continued fraction corresponds to a curve, parameterized by r, that is birationally equivalent to E1. This is stated without proof and is used to claim that all g_r have the same Hankel transform and hence the same 'essentially' EDS. If the main identification is established, this step also needs a justification or a precise statement of the birational equivalence used.
minor comments (4)
- [§2, Example 1] In the first displayed matrix, the row '−59 69 −4318 −5 1 0' appears to be a typo for '−59, 69, −43, 18, −5, 1, 0'; please check the spacing and values.
- [§4] The Hankel transform for the pseudo-involution cases is said to be the periodic sequence A010892 beginning 0, -1, -1, 0, 1, 1, ...; since a Hankel transform conventionally begins with the determinant of the 1x1 matrix, namely u_0 = 1, the indexing or the displayed initial terms should be clarified.
- [§3] The formula for u_n involves C_k; it would help to state explicitly that C_k is the k-th Catalan number and to define the ranges of the summation indices, as the multiple sums are otherwise hard to parse.
- [§2, Example 1] The continued fraction expressions use the notation [nP]_i for coordinates of multiples of P, but this notation is introduced only in the sentence immediately following; consider defining it at first use.
Circularity Check
No significant circularity: the Hankel transform is computed independently, and the asserted match with the elliptic divisibility sequence, though unproved, is not an input to the construction.
full rationale
The paper's chain is: elliptic curve, solve for y, discard initial terms, revert a modified generating function, obtain g(x), read off a Riordan array and lattice-path step set, then compute the Hankel transform of g(x). Each of these steps is performed algebraically from the curve parameters; the Hankel transform is computed from the resulting power series and is not fitted to the elliptic divisibility sequence. The elliptic divisibility sequence is introduced from the classical theory of the curve via refs. [7] and [21], and the comparison with the Hankel transform is made after both objects have been obtained independently. The appendix's Somos-4 statement is cited to independent references [5] and [21], not to the conclusion. The Riordan-array and lattice-path identifications are verified in the text by explicit algebraic identities such as u/x = 1 + (2(c-1)-a)u + (a(c-1)-b-(c-1)^2)ux + (a-2c+1)x + xu^2, whose solution is u = f(x). Self-references [2] and [3] are peripheral to the main derivation. The central sentence, 'The Hankel transform of u_n begins 1, ac-b-c^2, ... This is a (1, -ac+b+c^2) Somos 4 sequence, which essentially coincides with the elliptic divisibility sequence,' is an assertion without a proof or a precise normalization, and the index shift and sign conventions connecting the two sequences are never stated. That is a genuine mathematical gap and a correctness risk, but it is not circularity: no construction step is defined in terms of the alleged coincidence, and no parameter is chosen to force the equality. The apparent sign inconsistency between the front factor 1 + (a-2c+1)x and the Catalan argument 1 - x(a-2c+1) in the general formula for g(x) is likewise a correctness/typographical issue, not a circular reduction of the claimed result to its inputs.
Assumptions & free parameters
assumptions (4)
- domain assumption The A-matrix characterization of Riordan arrays and the resulting form of g(x) are valid.
- domain assumption The Hankel transform of (1+γx)/(1-αx-βx^2) C(δ x^3 (1+γx)/(1-αx-βx^2)^2) is a (δ^2, δ^2(αγ-β+γ^2)) Somos 4 sequence.
- standard math Elliptic divisibility sequences are Somos 4 sequences with parameters related to the curve.
- standard math Hankel transforms are invariant under binomial transforms.
invented entities (1)
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Signed lattice paths (negative multiplicities)
Cite this review
Pith. "Pith review of Elliptic Curves, Riordan arrays and Lattice Paths." pith.science (2026). https://pith.science/paper/BLWDN73D
@misc{pith2026250716765,
author = {Pith},
title = {Pith review of: Elliptic Curves, Riordan arrays and Lattice Paths},
year = {2026},
howpublished = {\url{https://pith.science/paper/BLWDN73D}},
note = {Machine review of arXiv:2507.16765}
}
abstract
In this note, we show that to each elliptic curve of the form $$y^2-axy-y=x^3-bx^2-cx,$$ we can associate a family of lattice paths whose step set is determined by the parameters of the elliptic curve. The enumeration of these lattice paths is by means of an associated Riordan array. The curves and the paths have associated Somos $4$ sequences which are essentially the same. For the curves the link to Somos $4$ sequences is a classical result, via the elliptic divisibility sequence. For the paths the link is via a Hankel transform.
Figures
Figures from the paper (7 more)
Reference graph
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Reviewed August 6, 2026 · model on record in the stance chip above.
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