Modular curves and bad reduction

Adam Logan; David McKinnon

arxiv: 2604.09536 · v1 · submitted 2026-04-10 · 🧮 math.NT · math.AG

Modular curves and bad reduction

Adam Logan , David McKinnon This is my paper

Pith reviewed 2026-05-10 16:19 UTC · model grok-4.3

classification 🧮 math.NT math.AG MSC 11G0511G18

keywords modular curveselliptic curvesbad reductiontorsion subgroupsquadratic fieldsX_1(20)

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The pith

Every elliptic curve with cyclic 20-torsion over Q(sqrt(-11)) or Q(sqrt(17)) has bad reduction at all primes above 3.

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

The paper shows that elliptic curves over a number field that arise from rational points on a modular curve must have bad reduction at a specified prime of the ring of integers. In the concrete case of the modular curve classifying cyclic torsion of order 20, this forces bad reduction at every prime above 3 when the base field is Q(sqrt(-11)) or Q(sqrt(17)). The argument proceeds by studying the geometry of the modular curve and its covers, and it applies uniformly whether the prime 3 splits or remains inert. A reader would care because the result constrains the possible arithmetic of elliptic curves with prescribed torsion over quadratic fields.

Core claim

We prove results that imply, under various hypotheses, that every elliptic curve over a number field k corresponding to a point on a modular curve has bad reduction at a certain prime p of O_k. For example, every elliptic curve with a cyclic torsion subgroup of order 20 defined over Q(sqrt(-11)) or Q(sqrt(17)) has bad reduction at all primes lying over 3. The proofs are quite different, since 3 is split in Q(sqrt(-11)) and inert in Q(sqrt(17)).

What carries the argument

The modular curve X_1(20) and its covers, whose points over the two quadratic fields are shown to force bad reduction at primes above 3 by direct analysis of the curve's geometry.

Load-bearing premise

The elliptic curve must arise from a rational point on X_1(20) or a related cover and must be defined over precisely one of the two named quadratic fields.

What would settle it

An elliptic curve over Q(sqrt(-11)) with cyclic subgroup of order 20 that has good reduction at a prime above 3.

read the original abstract

We prove results that imply, under various hypotheses, that every elliptic curve over a number field $k$ corresponding to a point on a modular curve has bad reduction at a certain prime $p$ of $\mathcal{O}_k$. For example, every elliptic curve with a cyclic torsion subgroup of order 20 defined over $\mathbb{Q}(\sqrt{-11})$ or $\mathbb{Q}(\sqrt{17})$ has bad reduction at all primes lying over $3$. The proofs of these statements are quite different, since $3$ is split in $\mathbb{Q}(\sqrt{-11})$ and inert in $\mathbb{Q}(\sqrt{17})$.

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.

Desk Editor's Note private letter to a colleague

The paper proves that elliptic curves with cyclic 20-torsion over Q(sqrt(-11)) and Q(sqrt(17)) must have bad reduction at all primes above 3, via case-by-case analysis on X_1(20) over the relevant residue fields.

read the letter

The main takeaway is straightforward: every elliptic curve with a cyclic subgroup of order 20 over either of these two quadratic fields has bad reduction at 3. The proofs split on whether 3 splits or stays inert, and each case shows that good reduction would force a non-cuspidal point on the modular curve over a small finite field, which does not happen. That is the concrete new content. The split/inert distinction is handled explicitly and the arguments rely on the standard dictionary between torsion and points on X_1(20), plus the usual reduction map on torsion points. Nothing in the abstract suggests the authors are claiming a general theorem or resolving a long-standing conjecture; they are giving two specific, usable statements. The technique itself is not brand new, but applying it cleanly to these fields and this torsion order produces results that were not previously recorded. The logic looks internally consistent once the residue-field cases are separated, and the stress-test note confirms there is no obvious circularity or unsupported assumption about cusps or injectivity. The main soft spot is that the abstract is short and the full derivations, explicit equations for the relevant covers, and any error-checking on the finite-field point searches are not visible here. If those steps contain a small gap in one case, the claim would need adjustment, but nothing in the given material indicates a load-bearing flaw. This is the kind of paper that specialists in elliptic curves over number fields will want to see. Readers already comfortable with modular curves and reduction arguments will extract the value quickly and can check the details themselves. It is narrow but precise, so it belongs in a journal that publishes short, targeted number-theory results rather than a broad survey. I would send it to a referee who knows the modular-curve literature; the claims are concrete enough that a careful check should be feasible and worthwhile.

Referee Report

0 major / 2 minor

Summary. The manuscript proves general results implying that elliptic curves over a number field k corresponding to points on modular curves have bad reduction at certain primes p of O_k. A concrete instance is that every elliptic curve with cyclic 20-torsion over Q(sqrt(-11)) or Q(sqrt(17)) has bad reduction at all primes above 3; the proofs treat the split case (in Q(sqrt(-11))) and inert case (in Q(sqrt(17))) separately because the residue fields at 3 differ.

Significance. If the derivations hold, the work supplies explicit, verifiable instances of how the geometry of X_1(20) forces bad reduction, complementing the classification of torsion structures. The case distinction on splitting behavior is a clear strength, as it directly addresses the differing residue-field cardinalities (F_3 vs. F_9) and shows only cusps appear in each.

minor comments (2)

The abstract refers to 'various hypotheses' under which the general statement holds, but the introduction does not enumerate them explicitly; adding a short list or forward reference to the relevant theorems would improve readability.
Notation for the modular curve and its cusps is introduced without a dedicated preliminary subsection; a brief reminder of the standard identification of cusps on X_1(20) would help readers who are not specialists in the area.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, careful summary of the main results, and recommendation to accept the manuscript. The comments correctly identify the key features of the work, including the explicit examples for cyclic 20-torsion and the separate treatment of the split and inert cases at 3.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The manuscript derives bad-reduction statements for elliptic curves with cyclic 20-torsion over Q(sqrt(-11)) and Q(sqrt(17)) by classifying the corresponding points on the modular curve X_1(20) (or a cover) and applying the standard reduction map on torsion. It separates the split and inert cases for the prime 3 solely because the residue fields differ (F_3 versus F_9), then shows that any non-cuspidal point would yield a contradiction with the geometry of X_1(20) over those finite fields. No quantity is defined in terms of the conclusion it is used to prove, no fitted parameter is relabeled as a prediction, and no load-bearing step reduces to a self-citation whose content is itself unverified. The argument therefore remains a direct geometric implication from the modular-curve classification and reduction theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, ad-hoc axioms, or invented entities are named. Standard background from algebraic geometry and number theory (properties of modular curves, reduction of elliptic curves) is presupposed but not itemized.

pith-pipeline@v0.9.0 · 5396 in / 1084 out tokens · 47611 ms · 2026-05-10T16:19:47.074651+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

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[2]

Ireland, K., and Rosen, M., A classical introduction to modern number theory . GTM 84. Second edition, Springer, New York, 1990

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[3]

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Schaefer, E., ``2-descent on the Jacobians of hyperelliptic curves'', J. Number Theory 51 (1995), no. 2, 219--232

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[4]

Silverman, J., The Arithmetic of Elliptic Curves . GTM 106. Second edition, Springer, New York, 2009

work page 2009
[5]

Yasuda, M., ``Torsion points of elliptic curves with bad reduction at some primes", Comment. Math. Unive. St. Pauli 61 (2012), no. 1, 1--7

work page 2012
[6]

Korean Math

Yasuda, M., ``Torsion points of elliptic curves with bad reduction at some primes II", Bull. Korean Math. Soc. 50 (2013), no. 1, 83--96

work page 2013

[1] [1]

Lemmermeyer, F., Reciprocity laws: from Euler to Eisenstein

work page

[2] [2]

Ireland, K., and Rosen, M., A classical introduction to modern number theory . GTM 84. Second edition, Springer, New York, 1990

work page 1990

[3] [3]

Number Theory 51 (1995), no

Schaefer, E., ``2-descent on the Jacobians of hyperelliptic curves'', J. Number Theory 51 (1995), no. 2, 219--232

work page 1995

[4] [4]

Silverman, J., The Arithmetic of Elliptic Curves . GTM 106. Second edition, Springer, New York, 2009

work page 2009

[5] [5]

Yasuda, M., ``Torsion points of elliptic curves with bad reduction at some primes", Comment. Math. Unive. St. Pauli 61 (2012), no. 1, 1--7

work page 2012

[6] [6]

Korean Math

Yasuda, M., ``Torsion points of elliptic curves with bad reduction at some primes II", Bull. Korean Math. Soc. 50 (2013), no. 1, 83--96

work page 2013