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arxiv: 1907.02737 · v1 · pith:C5YWEURKnew · submitted 2019-07-05 · 🧮 math.NT · math.AG· math.LO

Independence of CM points in Elliptic Curves

Jonathan Pila , Jacob Tsimerman This is my paper

Pith reviewed 2026-05-25 02:12 UTC · model grok-4.3

classification 🧮 math.NT math.AGmath.LO
keywords CM pointselliptic curveslinear dependenciesmodular curvesShimura curvesspecial pointscorrespondences
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The pith

All linear dependencies of CM points on elliptic curves are described

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

The paper establishes a complete description of the linear relations that hold among any number of points on elliptic curves obtained as images of CM points from modular or Shimura curves via fixed parameterizations. A reader would care because this controls the possible additive structures involving these arithmetic special points and provides a uniform framework that covers previous partial results. The description applies for each n separately and accounts for all such dependencies.

Core claim

The central claim is that for each n at least 1, there is an explicit description of all linear dependencies among n images in elliptic curves of special CM points coming from modular or Shimura curves under given parameterizations or correspondences. This unifies and improves upon earlier results in certain aspects.

What carries the argument

The fixed parameterizations or correspondences mapping CM points from modular or Shimura curves to points on elliptic curves.

If this is right

  • The description applies uniformly to any finite number n of such points.
  • Previous partial classifications are subsumed and extended in some cases.
  • Only the dependencies that arise from the geometry of the source curves occur.

Where Pith is reading between the lines

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

  • This classification could be applied to determine independence in concrete instances of maps and points.
  • It may connect to broader questions about the distribution of special points in arithmetic geometry.
  • Testable by checking small n cases against known examples from prior work.

Load-bearing premise

The parameterizations from the modular or Shimura curves to the elliptic curves are fixed independently of the CM points chosen.

What would settle it

An explicit set of n CM points whose images satisfy an unexpected linear relation not included in the described list would disprove the result.

read the original abstract

We prove a result which describes, for each $n\ge 1$, all linear dependencies among $n$ images in elliptic curves of special points in modular or Shimura curves under parameterizations (or correspondences). Our result unifies and improves in certain aspects previous work of Rosen-Silverman--K\"uhne and Buium-Poonen.

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 / 3 minor

Summary. The manuscript proves a result that, for each integer n ≥ 1, describes all linear dependencies among the n images (in elliptic curves) of special points lying on modular or Shimura curves, where the images are obtained via fixed parameterizations or correspondences. The result is presented as unifying and improving upon the earlier theorems of Rosen–Silverman–Kühne and Buium–Poonen.

Significance. If correct, the theorem supplies a uniform description of the linear relations satisfied by images of CM points under the indicated maps. This would consolidate two previously independent lines of work into a single statement and could serve as a reference point for further questions on heights or ranks of CM points in elliptic curves.

minor comments (3)
  1. The abstract states the result for 'special points' but the introduction should explicitly recall the precise definition of CM points on the source Shimura varieties that is used throughout the paper.
  2. Notation for the target elliptic curves and the parameterizations should be introduced once in §1 and then used consistently; several ad-hoc symbols appear in the statements of the main theorems.
  3. The comparison with the cited works of Rosen–Silverman–Kühne and Buium–Poonen would be clearer if a short table or paragraph listed the precise improvements (e.g., removal of a height bound, extension to higher-dimensional Shimura varieties).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper states a theorem unifying and improving prior independent results by Rosen-Silverman--Kühne and Buium-Poonen on linear dependencies of CM point images under fixed parameterizations. No self-citations appear in the abstract or described claim, no parameters are fitted and relabeled as predictions, and no ansatz or uniqueness result is imported from the authors' own prior work. The derivation is presented as a self-contained mathematical proof against external benchmarks in the theory of Shimura varieties and elliptic curves.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5572 in / 913 out tokens · 21039 ms · 2026-05-25T02:12:17.874029+00:00 · methodology

discussion (0)

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Reference graph

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