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arxiv: 2301.08312 · v3 · submitted 2023-01-19 · 🧮 math.NT

Computing torsion for plane quartics without using height bounds

Raymond van Bommel This is my paper

Pith reviewed 2026-05-24 09:28 UTC · model grok-4.3

classification 🧮 math.NT
keywords plane quarticsJacobian torsionrational torsionreduction modulo primesnumber fieldsgenus three curvesMagma implementationtorsion points
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The pith

An algorithm computes the rational torsion subgroup of Jacobians of plane quartics over Q by matching upper bounds from reduction modulo primes with explicit searches for points over Q and small number fields.

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

The paper presents an algorithm that determines the rational torsion subgroup of the Jacobian of a plane quartic curve over the rationals without using height bounds. Upper bounds come from reduction modulo primes while lower bounds are obtained by searching for torsion points not only over Q but also over small-degree number fields until the two bounds agree. Both complex analytic methods and Chinese remainder theorem techniques are employed to locate candidate points. The approach has been implemented in Magma and succeeds provably on more than 98 percent of the 82240 curves in a large test dataset.

Core claim

The algorithm provably computes the rational torsion subgroup of the Jacobian of a plane quartic over Q without relying on height bounds. Instead, it finds upper bounds for the torsion subgroup using reduction modulo primes and searches for torsion points over Q and small number fields until the bounds coincide, using both complex analytic and Chinese remainder theorem based methods to find such points.

What carries the argument

Matching of upper bounds obtained from reduction modulo primes against explicit searches for torsion points over Q and small number fields until the bounds agree.

If this is right

  • The torsion subgroup can be determined provably for the great majority of plane quartics with a rational point.
  • The combination of analytic and algebraic techniques suffices to locate the required points in practice.
  • The method terminates for more than 98 percent of the curves in the Sutherland dataset of 82240 examples.
  • Both the modular reduction step and the field-search step are required for the bounds to meet.

Where Pith is reading between the lines

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

  • The same matching strategy could be tested on other families of genus-3 curves once suitable modular reductions are available.
  • Curves on which the algorithm fails may systematically have torsion points of unusually high degree.
  • Knowing the torsion subgroup exactly is a prerequisite for computing the full Mordell-Weil group of the Jacobian.

Load-bearing premise

That the torsion subgroup is generated by points defined over number fields of degree small enough that the explicit search locates them before the modular upper bound is reached.

What would settle it

A plane quartic Jacobian over Q whose torsion subgroup requires a generator over a number field whose degree exceeds the search limit, so that the modular upper bound is never met by the points found.

read the original abstract

We describe an algorithm that provably computes the rational torsion subgroup of the Jacobian of a curve without relying on height bounds. Instead, the strategy is to find upper bounds for the torsion subgroup using reduction modulo primes, and searching for torsion points, not just over Q but also over small number fields, until the two bounds meet. Both complex analytic and Chinese remainder theorem based methods are used to find such torsion points. The method has been implemented in Magma for plane quartic curves over Q with a rational point and used to provably compute the rational torsion subgroup for more than 98% of Jacobians of curves in a data set due to Sutherland consisting of 82240 plane quartic curves.

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

1 major / 1 minor

Summary. The paper presents an algorithm to compute the rational torsion subgroup of the Jacobian of a plane quartic curve over Q without using height bounds. It obtains an upper bound on the torsion order as the lcm of #Jac(C)(F_p) over good primes p, then searches explicitly for torsion points of candidate orders over Q and over number fields of small degree (via complex-analytic and CRT methods) until the order of the subgroup generated by the discovered points matches the modular upper bound. The method is implemented in Magma for curves possessing a rational point and is reported to succeed in provably determining the torsion for more than 98% of the 82240 curves in Sutherland's dataset.

Significance. If the completeness argument holds, the work supplies a practical, height-bound-free method for determining rational torsion on genus-3 Jacobians, a task that is otherwise computationally delicate. The combination of modular upper bounds with targeted point searches over small extensions, together with the concrete implementation and large-scale verification on an existing dataset, would constitute a useful addition to the computational toolkit in arithmetic geometry.

major comments (1)
  1. [Abstract] Abstract: the claim that the algorithm 'provably computes' the full rational torsion subgroup rests on the assertion that explicit search over 'small number fields' will always locate a generating set whose order matches the modular upper bound. No derivation is supplied of an explicit, a-priori bound on the degree of the fields of definition of the torsion points in terms of the modular data (or any other computable invariant of C). Without such a bound the procedure is not guaranteed to terminate with a certified group; if a torsion point lies in an extension whose degree exceeds the search cutoff, the generated order remains strictly smaller than the upper bound and the algorithm does not certify completeness. This is load-bearing for the central claim of provability.
minor comments (1)
  1. A short table or paragraph enumerating the 2% of Sutherland curves on which the procedure failed to match bounds would help readers assess the practical reach of the search strategy.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The major comment raises a valid point about the scope of the provability claim, which we address below by clarifying the conditional nature of the certification. We will make targeted revisions to the abstract and introduction.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the algorithm 'provably computes' the full rational torsion subgroup rests on the assertion that explicit search over 'small number fields' will always locate a generating set whose order matches the modular upper bound. No derivation is supplied of an explicit, a-priori bound on the degree of the fields of definition of the torsion points in terms of the modular data (or any other computable invariant of C). Without such a bound the procedure is not guaranteed to terminate with a certified group; if a torsion point lies in an extension whose degree exceeds the search cutoff, the generated order remains strictly smaller than the upper bound and the algorithm does not certify completeness. This is load-bearing for the central claim of provability.

    Authors: We agree that no a priori bound on the degree of the fields of definition is derived or supplied, and that the procedure is therefore not guaranteed to terminate with a match for every curve. The algorithm certifies the full rational torsion subgroup precisely when the order of the subgroup generated by the discovered points equals the modular upper bound (which is independently proven to be an upper bound). In such cases the result is rigorous. The implementation searches over fields of bounded degree (via the described complex-analytic and CRT methods) and reports a provable computation only upon a match; the manuscript already states that this occurred for more than 98% of the curves in the dataset. We do not claim an unconditional decision procedure that always succeeds. We will revise the abstract and introduction to state explicitly that the algorithm provably determines the torsion subgroup conditional on the bounds matching within the searched range, and that success is not guaranteed for all curves. revision: partial

Circularity Check

0 steps flagged

No circularity: algorithm uses independent modular bounds and explicit search

full rationale

The paper presents an algorithm that obtains an upper bound on the torsion order from the lcm of #Jac(C)(F_p) over good primes and then performs explicit point searches over Q and bounded-degree extensions until the generated subgroup matches that order. This matching constitutes a proof for the cases where it occurs (98% of the tested set). No equations or steps reduce a claimed result to a fitted parameter or self-citation by construction; the modular bound and the search routines are independent of the final torsion subgroup. The description contains no self-definitional loops, no renaming of known results as new derivations, and no load-bearing self-citations. The limitation to 'small number fields' is an explicit practical cutoff rather than a hidden definitional assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on standard facts from algebraic geometry (Jacobians of smooth plane quartics are abelian varieties of dimension 3) and finite-group theory (torsion subgroups are finite). No free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (2)
  • standard math The Jacobian of a smooth plane quartic is an abelian variety of dimension 3 whose torsion subgroup is finite.
    Invoked implicitly when the paper speaks of computing the rational torsion subgroup.
  • standard math Reduction modulo primes yields valid upper bounds on the order of the torsion subgroup.
    Standard fact from the theory of abelian varieties over finite fields.

pith-pipeline@v0.9.0 · 5633 in / 1452 out tokens · 18810 ms · 2026-05-24T09:28:01.310444+00:00 · methodology

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Reference graph

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