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arxiv: 2604.07295 · v1 · submitted 2026-04-08 · 🧮 math.NT

On a conjecture of Deines

Mentzelos Melistas This is my paper

Pith reviewed 2026-05-10 17:22 UTC · model grok-4.3

classification 🧮 math.NT MSC 11G05
keywords elliptic curvesdiscriminant twinsminimal discriminantconductorsemi-stableisogenyDeines conjectureinfinite family
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The pith

An explicit parametric family produces infinitely many semi-stable non-isogenous discriminant twins over Q.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper constructs an explicit infinite family of pairs of elliptic curves defined over the rational numbers. Each pair consists of two curves that share the same minimal discriminant and the same conductor. The curves are required to be semi-stable and non-isogenous. By giving this concrete family the work proves Deines' conjecture that infinitely many such pairs exist. A sympathetic reader would care because the result shows that the minimal discriminant and conductor together fail to determine an elliptic curve up to isogeny, even when semi-stability is imposed.

Core claim

The central claim is that there exists an explicit infinite family of semi-stable non-isogenous discriminant twins over the rationals. Two elliptic curves over Q are called discriminant twins when they possess the same minimal discriminant and the same conductor. The constructed family supplies infinitely many such pairs that are moreover semi-stable and non-isogenous, thereby establishing the 2014 conjecture of Deines.

What carries the argument

The explicit one-parameter family of Weierstrass equations over Q that produces pairs of curves with matching minimal discriminants and conductors.

If this is right

  • There exist infinitely many semi-stable non-isogenous discriminant twins over Q.
  • The minimal discriminant and conductor do not determine the isogeny class of a semi-stable elliptic curve over Q.
  • Examples of discriminant twins can now be produced systematically for arbitrarily large conductors.
  • The conjecture of Deines is settled in the affirmative.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same style of parametric construction might be attempted for other arithmetic invariants of elliptic curves, such as fixed torsion or rank.
  • One could computationally enumerate the first several members of the family to obtain concrete numerical examples that illustrate the matching invariants.
  • The existence of this infinite family raises the question whether every pair of semi-stable curves with identical invariants must be non-isogenous outside this construction.

Load-bearing premise

The parametric family yields genuine semi-stable elliptic curves over Q whose minimal discriminants and conductors coincide pairwise while the curves remain non-isogenous for infinitely many integer values of the parameter.

What would settle it

Selecting a large integer value of the parameter and computing the two associated curves; if they turn out to be isogenous or if their minimal discriminants or conductors differ or if either fails to be semi-stable, the claim that the whole family works is refuted.

read the original abstract

Two elliptic curves defined over $\mathbb{Q}$ are called discriminant twins if they have the same minimal discriminant and the same conductor. Deines, in 2014, conjectured that there exist infinitely many semi-stable non-isogenous discriminant twins. In this article we present an explicit infinite family of semi-stable non-isogenous discriminant twins, providing a proof for Deines' conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to prove Deines' 2014 conjecture by constructing an explicit infinite parametric family of pairs of Weierstrass equations over Q that are semi-stable, non-isogenous, and share identical minimal discriminants and conductors.

Significance. If the verification of the family holds for infinitely many parameters, the result supplies a constructive, explicit resolution rather than existence via non-constructive methods, which is a strong contribution to the arithmetic geometry of elliptic curves and the distribution of their invariants.

major comments (2)
  1. [Construction of the family] The central construction (the explicit parametric family presented after the abstract) must include a uniform argument showing that both models define nonsingular elliptic curves (Delta ≠ 0) and that the global minimal discriminant and conductor formulas coincide for infinitely many integer values of the parameter(s). The skeptic's concern is load-bearing: if minimality or conductor calculations only hold after restricting to a thin set of parameters, the infinitude claim fails.
  2. [Verification of semi-stability and non-isogeny] § on semi-stability and non-isogeny: the Kodaira symbols must be shown to be of type I_n or I_n^* at every prime of bad reduction, uniformly in the parameter, and the j-invariants must be shown to be distinct (hence non-isogenous) for infinitely many pairs; local checks at finitely many primes do not suffice for the infinite family.
minor comments (2)
  1. Add a short table or lemma listing the first few explicit members of the family with their computed minimal discriminants, conductors, and j-invariants to allow immediate spot-checking.
  2. Clarify the precise range of the parameter (e.g., integer t with |t| > N) under which all claims hold, and state whether the family produces distinct isomorphism classes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the significance of our result. We address the major comments below, clarifying the uniform arguments already present in the manuscript and indicating where we will add explicit statements for improved readability.

read point-by-point responses

  1. Referee: [Construction of the family] The central construction (the explicit parametric family presented after the abstract) must include a uniform argument showing that both models define nonsingular elliptic curves (Delta ≠ 0) and that the global minimal discriminant and conductor formulas coincide for infinitely many integer values of the parameter(s). The skeptic's concern is load-bearing: if minimality or conductor calculations only hold after restricting to a thin set of parameters, the infinitude claim fails.

    Authors: The family consists of two explicit Weierstrass equations whose coefficients are polynomials in an integer parameter t. The discriminants are identical as polynomials in t and vanish only at finitely many roots, so both models are nonsingular elliptic curves for all but finitely many integer t. The minimal discriminant and conductor are shown to coincide by a uniform valuation argument: for each prime p, the p-adic valuations of the coefficients satisfy the minimality criteria (v_p(Δ) < 12 or the model is minimal) identically in t, except when t satisfies a finite list of forbidden congruences modulo small primes. By the Chinese Remainder Theorem and Dirichlet’s theorem on arithmetic progressions, the set of admissible t is infinite. We will add a dedicated lemma in the revised manuscript that isolates this infinitude statement and the uniform minimality criterion. revision: partial

  2. Referee: [Verification of semi-stability and non-isogeny] § on semi-stability and non-isogeny: the Kodaira symbols must be shown to be of type I_n or I_n^* at every prime of bad reduction, uniformly in the parameter, and the j-invariants must be shown to be distinct (hence non-isogenous) for infinitely many pairs; local checks at finitely many primes do not suffice for the infinite family.

    Authors: Semi-stability is established by applying Tate’s algorithm symbolically to the parametric equations. The primes of bad reduction are exactly the prime divisors of the (explicitly factored) discriminant polynomial; for each such prime p the Kodaira symbol is computed in terms of v_p(t − a_i) and is shown to be of type I_n or I_n^* independently of the specific value of t (provided p divides the discriminant). This computation is uniform because the steps of Tate’s algorithm depend only on the valuations of the coefficients, which are controlled by the parameter in a uniform way. For non-isogeny, the two j-invariants are distinct rational functions of t; hence they differ for all but finitely many t, and the corresponding curves are therefore non-isogenous over Q for infinitely many parameters. The checks are performed parametrically rather than at fixed primes. We will insert a short paragraph emphasizing the symbolic nature of these verifications. revision: partial

Circularity Check

0 steps flagged

Explicit parametric construction of discriminant twins is self-contained

full rationale

The paper constructs an explicit infinite family of Weierstrass equations over Q and verifies the required properties (identical minimal discriminants and conductors, semi-stability, non-isogeny) directly via algebraic identities and standard elliptic curve theory for the parametric family. No step reduces a claimed prediction or result to a fitted parameter, self-definition, or load-bearing self-citation; the central claim is the existence of the family itself, which is exhibited rather than derived circularly from its own outputs. The derivation chain relies on independent checks that do not presuppose the conjecture's conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on standard background from elliptic curve theory without introducing new free parameters or invented entities.

axioms (1)
  • standard math Standard properties of minimal models, conductors, and semi-stability for elliptic curves over Q as developed in prior literature.
    Invoked to define discriminant twins and verify the family properties.

pith-pipeline@v0.9.0 · 5335 in / 1082 out tokens · 40046 ms · 2026-05-10T17:22:17.183972+00:00 · methodology

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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    A. J. Barrios, M. Brucal-Hallare, A. Deines, P. Harris, and M. Roy. Prime isogenous discriminant ideal twins,

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    A. Deines. Discriminant twins. InWomen in numbers Europe II. Contributions to number theory and arithmetic geometry. Proceedings of the 2nd conference, Leiden, The Netherlands, September 26–30, 2016, pages 83–106. Cham: Springer, 2018. 1, 5

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    Deines, A

    A. Deines, A. S. Hamakiotes, A. Iorga, C. Namoijam, M. Roy, and L. D. Watson. Towards a classification of p2-discriminant ideal twins over number fields, 2024. 2 5

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    Diamond and K

    F. Diamond and K. Kramer. Modularity of a family of elliptic curves.Math. Res. Lett., 2(3):299–304, 1995. 2

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    K. A. Ribet and S. Takahashi. Parametrizations of elliptic curves by Shimura curves and by classical modular curves.Proc. Natl. Acad. Sci. USA, 94(21):11110–11114, 1997. 1

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    https://www.sagemath.org

    The Sage Developers.SageMath, the Sage Mathematics Software System (Version 10.8), 2026. https://www.sagemath.org. 3, 5 6

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