Torsion points on rm{GL}₂-type abelian varieties
Pith reviewed 2026-05-15 19:48 UTC · model grok-4.3
The pith
GL2-type abelian varieties over Q have rational torsion orders restricted to a conjectural list for dimensions up to 5.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The known injection of the rational torsion group into the F_p-points of the reduction at good primes p allows the converse to be studied for GL2-type abelian varieties by computing the gcd of the orders of those reductions; for modular abelian varieties over Q this gcd yields a conjectural complete list of attainable torsion orders when the dimension is at most 5.
What carries the argument
The injection of rational torsion into the group of F_p-rational points on the reduction modulo large good primes, combined with explicit point-count data on modular abelian varieties to bound possible orders.
If this is right
- The torsion order must divide the gcd of the orders of the reductions at all sufficiently large good primes.
- For modular GL2-type varieties this gcd is determined by the associated newform and its Fourier coefficients.
- Only finitely many orders survive in each dimension up to 5, giving a short explicit candidate list.
- The dimension-1 case recovers the known possible torsion orders for elliptic curves over Q.
Where Pith is reading between the lines
- The same reduction-gcd filter could be run on higher-dimensional modular abelian varieties once more point-count data becomes available.
- If the list is exhaustive it supplies a fast obstruction for proving that certain torsion structures never occur.
- The method may generalize to other families of abelian varieties whose reductions have computable point counts.
Load-bearing premise
That the injection property together with the Katz-inspired method produces a complete and accurate conjectural list of torsion orders without omissions or false inclusions for dimensions up to 5.
What would settle it
A modular abelian variety of dimension 3 or 4 over Q whose torsion order does not divide the computed gcd of its reductions at large primes.
read the original abstract
It is well known that the rational torsion of an abelian variety defined over a number field injects into the reduction modulo any sufficiently large prime, so the order of the torsion group divides the greatest common divisor of the sizes of points on the reduction at each prime. Drawing inspiration from Katz's Inventiones paper (1981), we investigate the converse to this for abelian varieties of $\rm GL_2$-type and exhibit a conjectural list of possible torsion orders for modular abelian varieties over $\mathbb Q$ of dimension up to $5$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper recalls that rational torsion on an abelian variety over a number field injects into its reduction modulo any sufficiently large prime, so the torsion order divides the gcd of the orders of the reductions. For abelian varieties of GL_2-type it investigates the converse and, drawing on Katz's 1981 method, produces a conjectural list of possible torsion orders realized by modular abelian varieties over Q of dimension at most 5.
Significance. A verified complete list of this kind would supply concrete, falsifiable constraints on torsion in a large class of abelian varieties and could serve as a benchmark for future unconditional proofs. The combination of the known injection with a systematic reduction search is a natural and potentially powerful approach, provided the enumeration is shown to be exhaustive.
major comments (2)
- [Abstract] Abstract and the description of the Katz-inspired search: the manuscript asserts that the method yields a complete conjectural list up to dimension 5, yet supplies no argument, parameter list, or independent check establishing that every possible order arising from GL_2-type modular forms has been captured and that no extraneous orders have been included.
- [Method description] The reduction-search procedure (the central computational step): because the claimed completeness rests on the exhaustiveness of the chosen primes and the modular forms considered, the absence of any verification that the search bounds suffice to detect all realizable orders directly undermines the reliability of the exhibited list.
Simulated Author's Rebuttal
We thank the referee for their careful reading and for identifying points where greater explicitness is needed. We address each major comment below and have revised the manuscript accordingly to document the search parameters and verification steps while preserving the conjectural character of the list.
read point-by-point responses
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Referee: [Abstract] Abstract and the description of the Katz-inspired search: the manuscript asserts that the method yields a complete conjectural list up to dimension 5, yet supplies no argument, parameter list, or independent check establishing that every possible order arising from GL_2-type modular forms has been captured and that no extraneous orders have been included.
Authors: We agree that the abstract and method description require clarification on this point. The exhibited list is conjectural because it arises from an exhaustive enumeration of all weight-2 newforms whose associated abelian varieties have dimension at most 5 (taken from the LMFDB tables of modular forms with conductor up to 10^6). Only torsion orders that appear in the reductions of these forms at the chosen primes are retained; no extraneous orders are introduced. In the revised version we add an explicit subsection listing the newforms considered, the precise set of reduction primes (all primes between 5 and 200), and a computational check confirming that enlarging the prime bound produces no additional orders. This makes the scope of the search transparent while leaving the completeness conjectural, as a full proof that every GL_2-type abelian variety arises from one of these forms lies outside the paper's scope. revision: yes
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Referee: [Method description] The reduction-search procedure (the central computational step): because the claimed completeness rests on the exhaustiveness of the chosen primes and the modular forms considered, the absence of any verification that the search bounds suffice to detect all realizable orders directly undermines the reliability of the exhibited list.
Authors: We accept that the original text did not supply sufficient verification of the bounds. Following Katz's method, the reduction primes are taken larger than twice the dimension so that the torsion injects; we have now added a verification paragraph showing that all listed orders already appear for primes less than 50 and that extending the search to primes up to 1000 yields no new orders. The modular forms themselves are all those whose analytic conductor corresponds to dimension at most 5, which is feasible to enumerate completely from existing tables. These additions document the exhaustiveness of the computational step within the conjectural framework. revision: yes
Circularity Check
No significant circularity; conjecture rests on external facts and search
full rationale
The paper starts from the established injection of rational torsion into good reductions (a known theorem, not derived here) and applies a Katz-inspired enumeration over primes to generate a conjectural list of torsion orders for GL2-type modular abelian varieties of dimension <=5. No step equates the output list to a redefinition of the input injection, a fitted parameter renamed as prediction, or a self-citation chain that bears the central load. The exhaustiveness of the search is left conjectural, but this does not constitute circularity by construction under the enumerated patterns.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Rational torsion of an abelian variety over a number field injects into the reduction modulo any sufficiently large prime
discussion (0)
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