A Constructive Proof of Masser's Theorem

Alexander J. Barrios

REVIEW 3 major objections 3 minor 12 references

A Constructive Proof of Masser's Theorem

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Four torsion types admit infinitely many good Frey curves, constructed from a recursive abc-triple process.

desk verdict A genuine constructive framework that fails on a false positivity lemma; the recursion leaves its domain for two of the four torsion subgroups. read the letter →

arxiv 1908.04938 v1 pith:NLFUQYI2 submitted 2019-08-14 math.NT

Alexander J. Barrios This is my paper · ORCID

classification math.NT MSC 11G05

keywords abcconjecturemodifiedSzpirogoodellipticcurvesFreytorsionsubgroupsrecursivetriplesmodularconstructiveproof

topics The abc Conjecture

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 gives a constructive proof of the classical theorem that infinitely many good elliptic curves exist, and it does so for each of the four rational torsion subgroups $C_2 \times C_{2N}$ with $N = 1,2,3,4$. A good elliptic curve here is one whose conductor is small compared with its $c_4$ and $c_6$ invariants, the objects controlled by the modified Szpiro conjecture, which is equivalent to the $abc$ conjecture. The construction starts from a single good $abc$ triple satisfying explicit congruences and a ratio threshold, then applies a polynomial recursion to produce an infinite sequence of good triples. Each resulting Frey curve is shown to be good and to have the prescribed torsion subgroup. If the construction is sound, it turns an existence theorem into explicit infinite families of examples.

What carries the argument

The central machinery is a table of four polynomial transformations $(A_T, B_T, C_T)$ together with the identities $A_T + B_T = C_T$ and a Bezout-type identity $U_T B_T + V_T C_T = W_T$ that controls coprimality. Each transformation is accompanied by a ratio function $f_T(t) = B_T(1,t)/A_T(1,t) - t$, which records how the ratio $b/a$ changes under one recursion step, and a positivity function $g_T(t)$ used to keep the resulting triple good. The torsion statement is carried by explicit universal elliptic curve models $X_t(T)$; an admissible change of variables identifies the Frey curve built from the transformed triple with $X_{t_T}(T)$, forcing the rational torsion subgroup to be exactly $T$. The congruences preserved by the recursion, such as $a_j \equiv 0 \pmod{16}$ and $b_j \equiv 1 \pmod 4$, make the resulting Frey curves semistable.

What would settle it

Evaluate $f_T(t)$ at $t = 3/2$ for $T = C_2 \times C_2$: with $f_T(t) = \frac{(1-t)^4}{8t(1+t^2)} - t$, the value is $\frac{1}{624} - \frac{3}{2} < 0$, while the table gives $\theta_T = 1$; this directly contradicts the positivity claim the recursion relies on. Recompute the first recursive triple for the paper's $C_2 \times C_2$ seed $P_0 = (25,72,3^4)$ and check whether $b_1/a_1$ remains above $\theta_T$; if it drops below, Proposition 3.4's conclusion fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that for each admissible torsion type $T = C_2 \times C_{2N}$, a recursive map on $abc$ triples produces an infinite family of good Frey curves. Starting from a good triple $P_0 = (a_0,b_0,c_0)$ with $a_0$ even, $b_0 \equiv 1 \pmod 4$, $b_0/a_0 > \theta_T$, and additionally $a_0 \equiv 0 \pmod 3$ when $T = C_2 \times C_6$, the recursion $P_j = (A_T(a_{j-1},b_{j-1}), B_T(a_{j-1},b_{j-1}), C_T(a_{j-1},b_{j-1}))$ yields, for every $j$, a good $abc$ triple whose induced Frey curve $y^2 = x(x-a_j)(x+b_j)$ is semistable, has torsion subgroup exactly $T$, and satisfies the good-curve inequality $N_E^6 < \max\{|c_4|^3, c_6^2\}$. This establishes Theorem 6.3, the constructive form of the infinitude theorem, uniformly for all four torsion types.

Load-bearing premise

The recursion stays inside its domain because, for every ratio above the stated threshold $\theta_T$, certain positivity inequalities (Lemma 2.1(6)) hold; for $T = C_2 \times C_2$ and $T = C_2 \times C_4$ that positivity assertion is false, so the step from $P_0$ to $P_1$ can land below the threshold.

Editorial extensions

If this is right

For each of the four torsion types, there are explicit infinite sequences of good $abc$ triples; for example, $T = C_2 \times C_2$ and $T = C_2 \times C_4$ both start from the seed $(25, 72, 3^4)$.
Every curve in these families is semistable and has modified Szpiro ratio strictly greater than 6, so the families are concrete witnesses to the infinitude that the modified Szpiro conjecture would forbid in the limit $\epsilon \to 0$.
The recursion preserves the congruence invariants needed for semistability at every step, so the construction is uniform across all four torsion types rather than requiring a separate argument for each.
Because the $abc$ conjecture and the modified Szpiro conjecture are equivalent, each infinite family of good Frey curves corresponds to infinitely many $abc$ triples with $\operatorname{rad}(abc) < c$, extending the phenomenon of the classical $(1, 9^k - 1, 9^k)$ family.

Reading between the lines

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

The recursion can be viewed as a dynamical system on ratios $r_j = b_j/a_j$ governed by $r_{j+1} = B_T(1,r_j)/A_T(1,r_j)$; if the positivity threshold issue were repaired, one could study growth rates, heights, and whether the ratios converge to $\theta_T$, yielding quantitative control on the quality decay visible in the computed examples.
The same template may extend to prescribed torsion structures over number fields, since the universal-curve step is what fixes the torsion; the rational classification restricts $T$, but the polynomial recursion itself does not obviously depend on that restriction.
The first few computed modified Szpiro ratios decay toward 6, which suggests that these families do not force a uniform gap above 6; one could test whether any uniform lower bound above 6 can be achieved infinitely often by similar recursive constructions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The paper is a genuine attempt to make Masser's theorem constructive, and its uniform treatment of the four possible torsion subgroups is a nice idea. The author builds explicit polynomial families AT, BT, CT from modular curve parameterizations, then defines a recursion on good ABC triples. If the technical inequalities worked, the recursion would produce infinitely many good Frey curves with prescribed torsion. That is a real contribution, and the paper is honest: it cites Masser, the modular curve sources, and its own earlier work for two cases.

The problem is load-bearing. Lemma 2.1(6) claims f_T(t), g_T(t), and the four polynomials are positive for t > theta_T. That is false. For T = C2 x C2, f_T(t) = ((1-t)^4)/(8t(1+t^2)) - t, and with theta_T listed as 1, f_T(2) = 1/80 - 2 < 0. For T = C2 x C4, f_T(2) = 9/16 - 2 < 0. So the lemma is simply wrong for two of the four cases. This is not a minor misprint: Lemma 3.2 uses the positivity of f_T to infer that b1/a1 > theta_T, and Proposition 3.4, Lemma 6.2, and Theorem 6.3 all depend on that step. The author's own example in Example 5.2 shows the failure: starting from P0 = (32,49,81), the recursion for C2 x C2 gives b1/a1 about 0.002, which is far below theta_T = 1. So the recursion leaves the domain on the very first step.

The rest of the paper is well-organized and the examples are clearly computed. There are no fitted parameters or invented entities, and the citations look appropriate. But the central induction does not hold as written. The constructive idea may be salvageable if the sign in the positivity statement is reversed or the domain of t is restricted correctly, but that would require rechecking the tables and likely changing the starting triples. As it stands, the proof of the main theorem is unsound.

I would not cite this in its current form, and it would not make my reading group's shortlist as a positive result. It could serve as a cautionary tale, though, and it deserves a serious referee who can check whether the lemma can be repaired. My recommendation: send it to review, and tell the author to fix the positivity claims and re-verify the induction.

Best, [You]

Referee Report

3 major / 3 minor

Summary. The manuscript presents a constructive proof of Masser's theorem that there are infinitely many good Frey curves, and it aims to refine the statement to each of the four possible torsion subgroups C2 x C2N with N = 1, 2, 3, 4. The proof defines rational maps given by the polynomials in Table 6, uses them to construct recursive ABC triples (Proposition 3.4), and then attaches Frey curves whose torsion and modified Szpiro ratio are controlled (Theorem 6.3). The paper also provides numerical examples and tables of the resulting qualities and Szpiro ratios.

Significance. If the construction were sound, the paper would give explicit infinite families of good elliptic curves for each of the four torsion subgroups allowed by Mazur's theorem, which is a concrete strengthening of Masser's existence result and a useful source of test cases for the abc/Szpiro conjectures. The paper also clearly separates the algebraic identities from the arithmetic inequalities and includes explicit starting triples. However, the main induction rests on a positivity claim that is false for two of the four torsion subgroups, so the central theorem is not established.

major comments (3)

[Lemma 2.1(6)] Lemma 2.1(6) asserts that f_T(t), g_T(t), A_T(1,t), B_T(1,t), C_T(1,t), D_T(1,t) are all positive for t > theta_T. This is false for T = C2 x C2 and T = C2 x C4. From Table 6, for T = C2 x C2 we have f_T(t) = (1-t)^4/(8t(1+t^2)) - t, so f_T(2) = 1/80 - 2 = -159/80 < 0 even though theta_T = 1. Similarly, for T = C2 x C4, f_T(t) = (1-t^2)^2/(4t^2) - t, so f_T(2) = 9/16 - 2 = -23/16 < 0. In fact f_T(1) = -1 for both cases, so theta_T = 1 is not even a root of f_T. Since Lemma 3.2 and all subsequent results use this positivity, the claimed positivity is load-bearing and false.
[Lemma 3.2, Proposition 3.4] Lemma 3.2 relies on Lemma 2.1(6) to infer that f_T(b/a) > 0 and hence B_T/A_T > b/a > theta_T. Because Lemma 2.1(6) is false for T = C2 x C2 and C2 x C4, the proof does not establish the key inequality B_T/A_T > theta_T that Proposition 3.4 needs to keep the recursion in its domain. The paper's own Example 5.2 for T = C2 x C2 starting from P0 = (32, 49, 81) gives b1/a1 = (a-b)^4/(8ab(a^2+b^2)) = 83521/42963200, approximately 0.00194, which is far below theta_T = 1 and contradicts the conclusion of Proposition 3.4 for j = 1. Thus the inductive step fails on the very example the paper presents.
[Theorem 6.3 and Lemma 6.2] Theorem 6.3 depends on Proposition 3.4 to ensure that every triple P^T_j satisfies b_j/a_j > theta_T, a_j even, b_j congruent to 1 mod 4, and the extra condition for T = C2 x C6. Since Proposition 3.4 is not valid for T = C2 x C2 and T = C2 x C4, the proof of Theorem 6.3 does not go through for those two torsion subgroups. The numerical examples in Table 4 likewise cannot be produced by the stated recursion without leaving the hypothesized domain, since the first step already violates b_1/a_1 > theta_T for T = C2 x C2.

minor comments (3)

[Example 5.2 / Table 2] The numerical values in Table 2 for T = C2 x C2 appear inconsistent with the definitions in Table 6: using a0 = 32 and b0 = 49, the recursion gives a1 = 8*32*49*(32^2+49^2) = 42963200 and b1 = (32-49)^4 = 83521, whereas the table lists a1 = 2511214657 and b1 = 38134. Please clarify whether the table entries are scaled or whether a different recursion was used.
[Lemma 2.1(7)] The statement of Lemma 2.1(7) is vacuously limited to T = C2 x C2N for N = 1,2, but the text labels it as part of Lemma 2.1; it may be clearer to present this as a separate remark or to justify why N = 3,4 are excluded.
[References] The paper cites [BTW10] for prior proofs of two cases, but that reference appears to be an unpublished MSRI Undergraduate Program report; the author does provide independent proofs here, yet the citation should be clearly marked as unpublished or replaced with a peer-reviewed source if one exists.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof is a forward algebraic recursion anchored in external modular-curve parameterizations; the sole self-citation is not load-bearing.

full rationale

The derivation chain starts from explicit polynomials A_T, B_T, C_T and verifies, by algebraic identities and radical estimates, that the recursively defined triples remain good ABC triples; the goodness of the associated Frey curves is then checked against conductor and discriminant inequalities. Nothing is fitted to the target conclusion. Lemma 4.3 proves the torsion statement by giving explicit Q-isomorphisms from the Frey curves to universal elliptic curves taken from HLP00 and Silverberg, then invoking Mazur's and Kamienny's torsion theorems; the C2 x C2 case is settled using Ono's external theorem. The self-citation [BTW10] after Theorem 4.4 records that two cases were previously handled by the author and collaborators, but the paper's own Lemma 4.3 supplies direct proofs for those cases, so the citation is not load-bearing. The reviewer's concern that Lemma 2.1(6) is false for C2 x C2 and C2 x C4 bears on the soundness of the positivity argument, not on circularity: if true, the lemmas would be genuine derivations, and if false, the proof fails for a factual reason rather than because a prediction was presupposed. Likewise, the assertion in Lemma 6.2 that c4,T(1,t)^3 - D_T(1,t)^6 is positive is an unproved analytic claim, not a definitional identification of output with input. There is no renamed known result and no ansatz whose only support is a self-citation. Hence the circularity score is 0.

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

The proof relies on standard torsion theorems and modular curve parameterizations from the literature. No free parameters are fitted to data, and no new entities are introduced. The computational positivity claims are not backed by machine-checkable artifacts.

assumptions (5)

standard math Mazur's Torsion Theorem: the possible rational torsion subgroups of an elliptic curve over Q are enumerated.
Used in Lemma 4.3 to conclude that a Frey curve with full 2-torsion and a point of order 2N has torsion exactly C2 x C2N.
standard math Kamienny's Torsion Theorem for elliptic curves over Q(i).
Used in Lemma 4.3, Case I, to rule out C2 x C8 over Q(i) and conclude the Q-torsion is C2 x C4.
domain assumption The universal elliptic curve parameterizations of Kubert, Howe-Leprevost-Poonen, and Silverberg have the stated torsion properties.
Lemma 4.1 relies on these parameterizations for the modular curves X1(2, 2N) and for the family W4.
domain assumption Ono's Main Theorem on concordant forms identifies the torsion of the model y^2 = x(x - 8ab(a^2+b^2))(x + (a-b)^4).
Used in Lemma 4.3, Case II, for T = C2 x C2.
standard math Fermat's right triangle theorem: the equation x^4 - y^4 = z^2 has no nontrivial integer solutions.
Used in Lemma 4.3, Case II, to rule out 8ab(a^2+b^2) being a square.

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Pith. "Pith review of A Constructive Proof of Masser's Theorem." pith.science (2026). https://pith.science/paper/NLFUQYI2

@misc{pith2026190804938,
  author       = {Pith},
  title        = {Pith review of: A Constructive Proof of Masser's Theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NLFUQYI2}},
  note         = {Machine review of arXiv:1908.04938}
}

abstract

The Modified Szpiro Conjecture, equivalent to the $abc$ Conjecture, states that for each $\epsilon>0$, there are finitely many rational elliptic curves satisfying $N_{E}^{6+\epsilon}<\max\!\left\{ \left\vert c_{4}^{3}\right\vert,c_{6}^{2}\right\} $ where $c_{4}$ and $c_{6}$ are the invariants associated to a minimal model of $E$ and $N_{E}$ is the conductor of $E$. We say $E$ is a good elliptic curve if $N_{E}^{6}<\max\!\left\{ \left\vert c_{4}^{3}\right\vert,c_{6}^{2}\right\} $. Masser showed that there are infinitely many good Frey curves. Here we give a constructive proof of this assertion.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 10 canonical work pages

[1]

Barrios, Caleb Tillman, and Charles Watts, Exceptional abc triples for frey curves with torsion subgroups z_ 2 z_ 4 and z_ 2 z_ 8 , MSRI Journal (2010)

Alexander J. Barrios, Caleb Tillman, and Charles Watts, Exceptional abc triples for frey curves with torsion subgroups z_ 2 z_ 4 and z_ 2 z_ 8 , MSRI Journal (2010)

work page 2010
[2]

Brian Conrad and Karl Rubin (eds.), Arithmetic algebraic geometry, IAS/Park City Mathematics Series, vol. 9, American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 2001, Including papers from the Graduate Summer School of the Institute for Advanced Study/Park City Mathematics Institute held in Park City, UT, June...

work page 2001
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Howe, Franck Lepr\'evost, and Bjorn Poonen, Large torsion subgroups of split J acobians of curves of genus two or three , Forum Math

Everett W. Howe, Franck Lepr\'evost, and Bjorn Poonen, Large torsion subgroups of split J acobians of curves of genus two or three , Forum Math. 12 (2000), no. 3, 315--364. 1748483

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[4]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Barrios, Caleb Tillman, and Charles Watts, Exceptional abc triples for frey curves with torsion subgroups z_ 2 z_ 4 and z_ 2 z_ 8 , MSRI Journal (2010)

Alexander J. Barrios, Caleb Tillman, and Charles Watts, Exceptional abc triples for frey curves with torsion subgroups z_ 2 z_ 4 and z_ 2 z_ 8 , MSRI Journal (2010)

work page 2010

[2] [2]

Brian Conrad and Karl Rubin (eds.), Arithmetic algebraic geometry, IAS/Park City Mathematics Series, vol. 9, American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 2001, Including papers from the Graduate Summer School of the Institute for Advanced Study/Park City Mathematics Institute held in Park City, UT, June...

work page 2001

[3] [3]

Howe, Franck Lepr\'evost, and Bjorn Poonen, Large torsion subgroups of split J acobians of curves of genus two or three , Forum Math

Everett W. Howe, Franck Lepr\'evost, and Bjorn Poonen, Large torsion subgroups of split J acobians of curves of genus two or three , Forum Math. 12 (2000), no. 3, 315--364. 1748483

work page 2000

[4] [4]

Kamienny, Torsion points on elliptic curves and q -coefficients of modular forms , Invent

S. Kamienny, Torsion points on elliptic curves and q -coefficients of modular forms , Invent. Math. 109 (1992), no. 2, 221--229. 1172689

1992

[5] [5]

London Math

Daniel Sion Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3) 33 (1976), no. 2, 193--237. 0434947

work page 1976

[6] [6]

D. W. Masser, Note on a conjecture of S zpiro , Ast\'erisque (1990), no. 183, 19--23, S\'eminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). 1065152

work page 1990

[7] [7]

Mazur, Modular curves and the E isenstein ideal , Inst

B. Mazur, Modular curves and the E isenstein ideal , Inst. Hautes \'Etudes Sci. Publ. Math. (1977), no. 47, 33--186 (1978). 488287

work page 1977

[8] [8]

161-162, Exp.\ No.\ 694, 4, 165--186 (1989), S\'eminaire Bourbaki, Vol

Joseph Oesterl\'e, Nouvelles approches du ``th\'eor\`eme'' de F ermat , Ast\'erisque (1988), no. 161-162, Exp.\ No.\ 694, 4, 165--186 (1989), S\'eminaire Bourbaki, Vol. 1987/88. 992208

work page 1988