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REVIEW 3 major objections 6 minor 33 references

Small Tamagawa numbers of elliptic curves with isogenies or torsion

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Pith's one-line read Cyclic rational isogenies of specified degrees restrict the global Tamagawa number of an elliptic curve over $\mathbb{Q}$ to a very small set of prime divisors, and infinite families attain the small values.

desk verdict Genuinely new classification of Tamagawa primes for rational isogenies, with two fixable gaps; deserves referee time. read the letter →

arxiv 2505.20479 v2 pith:ADYUEYHO submitted 2025-05-26 math.NT

Mentzelos Melistas This is my paper · ORCID
classification math.NT MSC 11G0511G0711G40
keywords TamagawanumbersellipticcurvesoverQrationalisogeniestorsionpointsreductiontypesquadratictwistssurfacesBSDformula
topics Birch and Swinnerton-Dyer
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

Tamagawa numbers measure, prime by prime, how an elliptic curve degenerates at bad primes; the global Tamagawa number $c(E)$ is their product. This paper proves that if $E/\mathbb{Q}$ has a cyclic $\mathbb{Q}$-rational isogeny of degree $N$ in a list of possible degrees, then $c(E)$'s prime divisors are severely restricted: for $N=14,19,43,67,163$ one gets $c(E)=2^n$; for $N=11,27,37$ one gets $2^n3^m$ with at most one factor $3$; for $N=17,21,15$ one extra prime appears at most to the first power; and for $N=9$ the power of $3$ is capped by $3^3$. The paper also shows that for rational torsion points of order $4,5,6,7,8,9,10,12$ no such restriction is possible, since every prime can be made to divide $c(E)$ for infinitely many curves. On the positive side it constructs infinite families with small Tamagawa numbers: order-$4$ torsion with $c(E)\in\{4,8,12\}$, order-$5$ torsion with $c(E)\le 30$, and isogenies of degree $5,7,13$ with $c(E)=2^n3^m$. These results matter because $c(E)$ is one of the factors in the BSD formula, so controlling it controls a global arithmetic invariant for whole families of curves.

What carries the argument

The carrying mechanism is the reduction-type dictionary for elliptic curves with $N$-isogenies. The modular curve $X_0(N)$ parametrizes pairs $(E,C)$ with a cyclic rational $N$-isogeny; for each $N$ in Theorem 1.1 its rational points give a finite list of $j$-invariants, and none are $0$ or $1728$, so every such curve is a quadratic twist of one of finitely many base curves. Two tables do the work: the table from the standard local reduction algorithm converting reduction types into local Tamagawa numbers (for example split $I_n$ gives $n$, non-split $I_n$ gives $1$ or $2$, $III$ gives $2$, additive types give $1$ or $3$), and the quadratic-twist table saying how reduction types swap at primes dividing the twist. For the specialization results, the mechanism is different: sieve theorems guarantee that the discriminant polynomial $\Delta(T)$ of a one-parameter family $E/\mathbb{Q}(T)$ takes values with few prime factors, so that specializations $E_n$ have only a few primes of bad reduction and $c(E_n)$ is bounded by an explicit expression in $\deg\Delta$.

What would settle it

Recompute, with the standard local reduction algorithm, the reduction types at $p=2$ and $p=3$ of the four degree-$21$ base curves of conductor $162$ and the four degree-$15$ base curves of conductor $50$; if the degree-$21$ types are not $(I_3,I_1,I_{21},I_7)$ at $p=2$ and $(II,II^*,II,II^*)$ at $p=3$, or the degree-$15$ types are not $(I_1,I_3,I_5,I_{15})$ at $p=2$, the corresponding part of Theorem 1.1 fails. A single curve with a rational $21$-isogeny and $c_3(E)>3$ or $c_2(E)>21$ would also disprove part (iv).

Watch

Extended reading notes

Core claim

The central claim of the paper is Theorem 1.1. Let $E/\mathbb{Q}$ be an elliptic curve with a cyclic $\mathbb{Q}$-rational isogeny of degree $N$. Then: for $N=14,19,43,67,163$, $c(E)=2^n$ with $n\ge1$; for $N=11,27,37$, $c(E)=2^n3^m$ with $m\in\{0,1\}$; for $N=17$, $c(E)=2^n3^m17^k$ with $m,k\in\{0,1\}$ and $n\ge1$; for $N=21$, $c(E)=2^n3^m7^k$ with $m\in\{0,1,2\}$, $k\in\{0,1\}$, $n\ge1$; and for $N=9$ or $15$, $c(E)=2^n3^m\ell^k$ with $m\in\{0,1,2\}$, $k\in\{0,1\}$, where $\ell=3$ or $5$ respectively. The proof shows that for each of these degrees every local Tamagawa factor $c_p(E)$ can only be $1,2,4$, with a single possible factor $3$, $5$, $7$, or $17$ in specified cases. It also proves that the remaining torsion orders $4,5,6,7,8,9,10,12$ admit no such prime bound, and supplies infinite families where $c(E)$ is as small as the constraints allow.

Load-bearing premise

The load-bearing premise is that the quoted reduction types of the small base curves at primes $2$, $3$, and $5$—how each curve degenerates modulo those primes—are correct as taken from the database; the paper does not recompute them, and a single error there would break the corresponding restriction in Theorem 1.1.

Editorial extensions

If this is right

  • For every curve with a rational isogeny of degree $19,43,67$, or $163$, the global Tamagawa number is a power of $2$, so the BSD quotient $c(E)/|E(\mathbb{Q})_{\mathrm{tors}}|$ has no odd primes in its Tamagawa part.
  • For isogeny degrees $11,27,37$ the $3$-part of $c(E)$ is at most $3$; for degrees $17,21,15$ the extra prime appears at most to the first power and the $3$-part at most $9$; for degree $9$ the $3$-part is at most $27$.
  • A rational $14$-isogeny forces $c(E)$ to be even, refining the earlier result that $2\mid c(E)$ for such curves.
  • Infinitely many $j$-distinct curves with isogeny degree $5,7$, or $13$ have $c(E)=2^n3^m$, and infinitely many curves with a rational point of order $4$ (respectively $5$) have $c(E)\in\{4,8,12\}$ (respectively $c(E)\le30$).
  • For a non-isotrivial one-parameter family of elliptic curves whose discriminant has few roots modulo every prime, infinitely many specializations satisfy the explicit bound $c(E_n)\le16(\log_2(m)+s\deg\Delta)^{d(m)+s}$, and under the $abc$-conjecture with squarefree discriminant values there are infinitely many specializations with $c(E_n)=1$.

Reading between the lines

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

  • The same finite-$j$-invariant plus quadratic-twist strategy could be applied to other isogeny degrees with a known table of $j$-invariants, such as degree $25$, to predict analogous prime restrictions; the paper does not attempt this.
  • The sieve bound in Theorem 1.3 depends on an explicit but non-optimal constant $s$ from the almost-prime theorem; a sharper sieve would shrink the bound and could make the difference $C_2-C_1$ in Question A depend only on $\deg\Delta$.
  • For most of the small torsion orders, Proposition 3.5 shows any prime can be forced into $c(E)$, which indicates that Theorem 1.1's restrictive list is close to optimal.
  • The parity formula $v_7(c(E))\equiv v_7(c(E'))+\operatorname{rk}E/\mathbb{Q}+1\pmod2$, conditional on the standard finiteness assumption for the relevant Galois cohomology group, offers a local-data route to rank-parity statements for curves with $7$-torsion and semistable reduction at $7$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper studies Tamagawa numbers c(E) of elliptic curves over Q that have rational cyclic isogenies or torsion points. Theorem 1.1 claims restrictions on the prime divisors of c(E) according to the degree N of a cyclic Q-rational isogeny, with N ranging over the Mazur-Kenku list. Section 2 proves these restrictions using reduction-type classifications from the author's earlier work [23] and quadratic-twist tables, and also produces infinite families with isogenies of degree 5, 7, and 13 whose Tamagawa numbers are of the form 2^n 3^m. Section 3 constructs infinite families with a rational point of order 4 or 5 and small Tamagawa numbers, and proves that for N'=4,5,6,7,8,9,10,12 any prime can divide c(E) for infinitely many curves with an N'-isogeny. Section 4 gives bounds on Tamagawa numbers of specializations of elliptic surfaces in terms of the discriminant of a minimal Weierstrass equation, including an abc-conditional result producing infinitely many specializations with Tamagawa number 1.

Significance. If the issues below are corrected, the paper makes a useful contribution to a natural question: which primes divide Tamagawa numbers of elliptic curves with prescribed rational isogenies, and how small can these Tamagawa numbers be. The paper uses standard tools (Tate's algorithm, quadratic-twist tables, the Mazur-Kenku classification, sieve results) in a transparent way, and several of the construction results in Sections 3 and 4 are concrete and unconditional. The main theorem's prime-divisor restrictions, if valid, would refine earlier work of Lorenzini and Trbović. The dependence on LMFDB local data is a legitimate gap in exposition, but it is likely fillable by direct Tate-algorithm computations. The paper is not circular: it relies on earlier published work [23] and external classifications.

major comments (3)
  1. The exponent condition n≥1 in Theorem 1.1(iv)-(v) is not justified and appears to be contradicted by the local data in the same paper. In Proposition 2.10, the base curve E2 = 50.a1 has reduction type I3 modulo 2 and IV modulo 5, and the proposition states that all curves in that isogeny class have conductor 2·5^2. Since the only bad primes are 2 and 5, the local Tamagawa numbers are c2∈{1,3} (odd, by Table 1 for I3) and c5∈{1,3} (odd, by Table 1 for IV). Thus the global c(E2) is odd, contradicting n≥1 in Theorem 1.1(v). Similarly, the curve 162.c2 named in Remark 2.13 has c2=21 and reduction type II at 3 (so c3=1), with conductor 2·3^4, giving c(E)=21, which has no factor 2 and contradicts n≥1 in Theorem 1.1(iv). Please correct the exponent range, most likely to n≥0, or provide a separate argument showing that these curves do not have the required global Tamagawa numbers.
  2. [Propositions 2.10 and 2.12] The reduction types of the base curves 50.a1-50.a4 at p=2,5 and 162.c1-162.c4 at p=2,3 are quoted from LMFDB [17] without an independent computation or a citable proof. These local data are load-bearing for Theorem 1.1(iv)-(v), because the twist argument transfers only these base types to all twists. Please include either a Tate-algorithm computation for each base curve, or explicit minimal Weierstrass equations together with the resulting Kodaira types, so that the proof is self-contained and the reader can verify the claimed values c2=1,3,5,15 for N=15 and c2=1,2,3,4,7,21 for N=21.
  3. [Proposition 2.16, proof] The proof contains the assertion "v_q(j(E_{t0,ℓ})) > 0 for every prime q ≠ p" after taking t0=p. This is false: for the rational function F_ℓ(t), with denominator t, one has v_q(j) ≥ 0 for all q ≠ p, with equality for most q. The desired bound c_q(E) ≤ 4 follows from [28, Corollary 9.2] already under v_q(j) ≥ 0, so the error is local and does not destroy the conclusion, but the inequality must be corrected to v_q(j) ≥ 0.
minor comments (6)
  1. [Proposition 2.14, proof] The first sentence of the proof refers to an isogeny of degree 14, but the proposition concerns degree 27; this is a typo.
  2. [Proposition 2.4, conclusion] The concluding claim "c(E)=2^n3^m, for some n≥2" does not follow from the listed possibilities (i)-(iv), since c2=1, c5=2, and all other local factors equal to 1 would give n=1. Please verify whether the intended bound is n≥1 or whether an additional source of a factor 2 is forced by [23, Theorem 3.7].
  3. [Proposition 2.1(i)] The wording "If p ̸= 2, ℓ is a prime" should read "If p is a prime different from 2 and ℓ".
  4. [Theorem 4.3, proof] The inequality "c_p(E_n) ≤ v_p(E_n)" should refer to v_p(Δ(n)), not v_p(E_n).
  5. [Remark 2.15] The sentence "Let q be a divisor of q" is a typo; it should refer to a prime divisor q of d.
  6. [Throughout] There are several grammatical slips (e.g., "we will show a slightly statement" in the proof of Proposition 3.5) and minor notation inconsistencies in Section 2; these should be cleaned up during revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's arguments reduce to external classifications, database local data, and independently proved prior theorems, not to their own conclusions.

full rationale

The paper's central results are obtained by combining external classifications (Mazur, Kenku, Lozano-Robledo's j-invariant lists), standard Tate algorithm/Tamagawa tables, and local reduction-type data for base curves. The author's earlier paper [23] is cited as the source of reduction-type computations for isogenies of degrees 11, 17, 19, 37, 43, 67, and 163; that prior work is itself a separate published derivation from explicit equations for X0(N), and it does not presuppose the present Tamagawa-number restrictions. For degrees 14, 15, 21, and 27, the proof uses j-invariants from [20, Table 4] and reduction types of LMFDB twist representatives (50.a1-50.a4 and 162.c1-162.c4), then applies quadratic-twist tables (Table 2) and Tamagawa tables (Table 1). These are inputs, not conclusions being assumed: the claimed restrictions on primes dividing c(E) are deduced from the possible Kodaira types, not baked into the definition of those types. The proof of Proposition 2.16 contains a minor verbal slip ('v_q(j)>0' where 'v_q(j) ≥ 0' is meant), but the bound c_q ≤ 4 follows either way, so it is a typo rather than a circular step. No equation or fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the author's own work to forbid alternatives, and no known result is merely relabeled. The reliance on LMFDB reduction-type data is a verification gap, not circularity, and it does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard results in arithmetic geometry (isogeny classification, Tate's algorithm, twist reduction rules) and on two unproved conjectures used only in explicitly conditional statements (finiteness of X(E/Q) and abc). No parameters are fitted to data, and no ad hoc objects are introduced. The paper's reliance on the author's own earlier work [23] is non-circular because those results are proved independently.

assumptions (6)
  • standard math Classification of possible degrees of Q-rational isogenies: N ≤ 19 or N ∈ {21,25,27,37,43,67,163} (Mazur, Kenku).
    Used to enumerate cases in Theorem 1.1; standard theorem cited from [15] and [21].
  • standard math Tate's algorithm and the dictionary between reduction types and Tamagawa numbers (Table 1).
    Used throughout to convert reduction types into values of c_p(E).
  • standard math Quadratic twist reduction rules (Comalada, Proposition 1, restated as Lemmas 2.6 and 2.7).
    Used in Propositions 2.8-2.14 to reduce general curves to the base curves of smallest conductor in each twist class.
  • domain assumption Finiteness of the Tate-Shafarevich group of E/Q (assumed in Proposition 2.18).
    Needed to apply the isogeny invariance of the BSD formula; this is a long-standing open conjecture and is explicitly flagged as an assumption in the proposition.
  • domain assumption The abc-conjecture (assumed in Theorem 4.3).
    Used via Granville's theorem [11] to ensure infinitely many n with Δ(n) squarefree; the result is explicitly conditional on abc.
  • standard math Sieve theorem of Halberstam and Richert (Theorem 4.1), providing existence of an explicit constant s.
    Used in the proof of Theorem 1.3 to find infinitely many n with F(n) ∈ P_s; this is a published theorem, though the constant s is not computed in the paper.

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Pith. "Pith review of Small Tamagawa numbers of elliptic curves with isogenies or torsion." pith.science (2026). https://pith.science/paper/ADYUEYHO

@misc{pith2026250520479,
  author       = {Pith},
  title        = {Pith review of: Small Tamagawa numbers of elliptic curves with isogenies or torsion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ADYUEYHO}},
  note         = {Machine review of arXiv:2505.20479}
}
abstract

In this article with study Tamagawa numbers of elliptic curves defined over $\mathbb{Q}$ that have isogenies or torsion points. More precisely, our aim is either to bound the set of primes primes that can divide their Tamagawa numbers or, when such a bound is not possible, to find infinite subfamilies whose Tamagawa numbers are as small as possible. Finally, we also investigate Tamagawa numbers of specializations of elliptic surfaces.

Discussion (0). Continue with ORCID to comment.

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