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arxiv: 1906.10580 · v1 · pith:Q6QT3FDXnew · submitted 2019-06-25 · 🧮 math.NT

Integrality properties in the Moduli Space of Elliptic Curves: CM Case

Stefan Schmid This is my paper

Pith reviewed 2026-05-25 16:08 UTC · model grok-4.3

classification 🧮 math.NT
keywords singular modulialgebraic unitsj-invariantscomplex multiplicationelliptic curvesmoduli spacefinitenessexplicit bounds
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The pith

For fixed non-CM algebraic α, only finitely many singular moduli j satisfy that j-α is an algebraic unit, with explicit bounds supplied.

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

The paper establishes a finiteness result for singular moduli in the moduli space of elliptic curves. Given a fixed j-invariant α of a non-CM elliptic curve, it shows there are only finitely many CM j-invariants j such that j minus α is an algebraic unit. This refines an earlier non-effective finiteness theorem by replacing equidistribution arguments with explicit bounds on the possible j. A reader would care because the result converts an existence statement into something that can be checked or used for concrete arithmetic computations involving CM points and units.

Core claim

For a fixed j-invariant α in the algebraic closure of the rationals coming from an elliptic curve without complex multiplication, there are only finitely many singular moduli j such that j-α is an algebraic unit. The proof supplies explicit bounds on the number or size of these singular moduli, in contrast to prior work that established finiteness without effectivity.

What carries the argument

Singular moduli (j-invariants of CM elliptic curves) together with the condition that their difference from a fixed non-CM α is an algebraic unit; the argument makes the finiteness effective by supplying explicit bounds.

If this is right

  • The collection of singular moduli satisfying the unit-difference condition is finite and its cardinality is bounded by an explicit constant depending on α.
  • The result applies specifically in the non-CM case for α and addresses the CM case for the varying j.
  • The bounds allow in principle for an algorithmic search to list all such singular moduli for a given α.
  • This provides an effective version of the integrality statement in the moduli space for differences involving one CM and one non-CM point.

Where Pith is reading between the lines

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

  • The explicit bounds could be combined with height estimates or class number bounds to produce fully computable lists for small-degree α.
  • Similar effective techniques might extend to other Diophantine conditions on differences of j-invariants beyond algebraic units.
  • The result suggests that arithmetic properties of the moduli space become more tractable once one point is fixed and non-CM.

Load-bearing premise

The fixed α must itself not be a CM j-invariant.

What would settle it

An explicit non-CM α together with an infinite sequence of distinct singular moduli j_n where each j_n - α is an algebraic unit, or a single such j whose size exceeds the paper's explicit bound.

Figures

Figures reproduced from arXiv: 1906.10580 by Stefan Schmid.

Figure 1
Figure 1. Figure 1: Application of the maximum modulus principle to the blue area. on the circle |τ − ξ| = δ. Also Im(τ ) ≥ 6 Im(ξ) implies Im(τ ) ≥ Im(ξ) + 1, so we obtain by applying Proposition 3.2 twice |j(ξ)| ≤ 2079 + e 2π Im(ξ) = 2079 − 20e 2π √ 3 2 + 20e 2π √ 3 2 + e 2π Im(ξ) < −2079 + 20e 2π √ 3 2 + e 2π Im(ξ) ≤ −2079 + 20e 2π Im(ξ) + e 2π Im(ξ) ≤ −2079 + 21e 2π Im(ξ) < −2079 + e 2π e 2π Im(ξ) ≤ −2079 + e 2π Im(τ) ≤ |… view at source ↗
Figure 2
Figure 2. Figure 2: Neighborhoods of points on the boundary Lemma 3.9 In the same setting as in the previous lemma, if |j(τ )−j(ξ)| < c(ξ), then |τ − Mξ| < δ with Mξ ∈ F¯ for some M ∈ T where T = { [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
read the original abstract

A result of Habegger shows that there are only finitely many singular moduli such that $j$ or $j-\alpha$ is an algebraic unit. The result uses Duke's Equidistribution Theorem and is thus not effective. For a fixed $j$-invariant $\alpha \in \bar{\mathbb{Q}}$ of an elliptic curve without complex multiplication, we prove that there are only finitely many singular moduli $j$ such that $j-\alpha$ is an algebraic unit. The difference to the work of Habegger is that we give explicit bounds.

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

0 major / 1 minor

Summary. The manuscript proves that for a fixed non-CM j-invariant α ∈ Q-bar, there are only finitely many singular moduli j such that j − α is an algebraic unit in the ring of integers of Q(j, α). Explicit (effective) bounds are given on the height/degree of such j, improving on Habegger's finiteness result by avoiding Duke's equidistribution theorem and deriving effectivity from an alternative height estimate.

Significance. If the central claim holds, the result strengthens the arithmetic theory of singular moduli by supplying effective bounds, which are useful for computational checks and Diophantine applications involving units in fields generated by CM j-invariants. The explicit nature of the bounds and the avoidance of non-effective equidistribution tools constitute a clear advance over prior work.

minor comments (1)
  1. [Abstract] Abstract: the phrase 'explicit bounds' is used without indicating their dependence on the minimal polynomial or height of α; a single sentence clarifying the form of the bound would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. We appreciate the recognition that our effective bounds improve on Habegger's result by avoiding non-effective equidistribution.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states a finiteness theorem with explicit bounds on CM j-invariants satisfying the unit condition for fixed non-CM α, explicitly contrasting its effectivity with Habegger's non-effective Duke-based argument. No equations, parameters, or premises are shown to reduce to fitted inputs, self-definitions, or load-bearing self-citations. The distinction from prior work is precisely the provision of an independent effective height or Diophantine estimate, making the central claim self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no access to the body of the paper, so no concrete free parameters, axioms, or invented entities can be extracted.

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Works this paper leans on

7 extracted references · 7 canonical work pages

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