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Prime order torsion on elliptic curves over number fields. Part I: Asymptotics

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Pith's one-line read Conditionally on sparse 'strange' newforms, the largest prime order of rational torsion on elliptic curves over degree-d fields is at most 3d+1 for large even d and o(d) for odd d.

arxiv 2505.14109 v1 pith:R6PQTNLY submitted 2025-05-20 math.NT math.AG

Maarten Derickx , Michael Stoll This is my paper · ORCID
classification math.NTmath.AG
keywords primeasymptoticscurvesellipticfieldsnumberanalyticassuming
verification ladder T0 review T1 audit T2 compute T3 formal

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The reading

An elliptic curve is a geometric object whose points form a group, and a number field is a way of extending rational arithmetic to higher-dimensional coordinates. For a field of degree d, one asks which prime numbers p can occur as the order of a torsion point on an elliptic curve. This set is written S(d). For small d the answer is known by hard computation and old theorems. This paper asks what happens as d grows.
Extended reading notes

Core claim

Theorem 9.1: Assume Conjectures 5.3(1) and 8.4(1). Fix m >= 1. Then there is d0(m) such that for integers d >= d0(m), (1) if d is odd, Snew(d) is contained in Primes(d/m); (2) if d is even, Snew(d) is contained in Primes(d/m) union {d/k+1 : 1 <= k <= m, k|d} union {2d+1, 3d+1}. If correct, Corollary 9.2 gives max S(d) <= 3d+1 for sufficiently large even d and max S(d) = o(d) for odd d.

Load-bearing premise

The load-bearing premise is Conjectures 5.3(1) and 8.4(1), which assert that strdim(p)/log p -> 0 and perdim(p)/log p -> 0 as p runs over primes. These are new, empirically motivated, unproved statements about the sparsity of newforms with unexpectedly high analytic rank. Theorem 7.7 depends on 5.3(1); Theorem 8.5 depends on 8.4(1); Theorem 9.1 uses both, specifically to force the degree of points on X0(p) to be even in the exceptional case. If either conjecture fails, the proof thresholds collapse.

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Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data. The central conditional statements rest on two new conjectures, Conjectures 5.3(1) and 8.4(1), about the sparsity of certain newforms. All other assumptions are standard theorems in the arithmetic of modular curves and Jacobians. Strange primes and perdim are definitional invariants of modular form data, not additional postulated entities.

assumptions (6)
  • ad hoc to paper Conjecture 5.3(1): strdim(p)/log p -> 0 as p runs over all primes.
    Newly introduced by this paper. Used in Theorem 7.7 and Theorem 9.1 to ensure p is larger than (2*sqrt(2)+3)^{strdim(p)} for large p.
  • ad hoc to paper Conjecture 8.4(1): perdim(p)/log p -> 0 as p runs over all primes.
    Newly introduced by this paper. Used through Theorem 8.5 in Theorem 9.1 to force the exceptional degree on X0(p) to be even.
  • standard math Yau-Abramovich-Kim-Sarnak gonality lower bound: gon_Q(X_H) >= (325/2^15) * (p^2 - 1)/(2#H).
    This proven bound supplies the quantitative gonality thresholds used throughout Sections 4 through 9.
  • standard math Kolyvagin-Logachev and Kato: for the relevant modular abelian varieties, analytic rank zero implies Mordell-Weil rank zero, and positive analytic rank is detected by modular symbols.
    Used in Sections 5 and 8 to identify positive analytic rank with positive Mordell-Weil rank and to justify the strange prime computations.
  • standard math Mazur's theorem: X0(p) has no non-cuspidal rational points for p > 163.
    Used in Remark 7.6 to exclude m=1 in the degree-m point on X0(p) in case (d).
  • standard math Hasse bound: for a weight 2 newform f, the eigenvalue a_2(f) satisfies |a_2(f)| <= 2*sqrt(2).
    Used in the proofs of Corollary 7.4 and Proposition 8.2 to bound the coefficients of the polynomial h whose roots are the a_2(f_j).

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Pith. "Pith review of Prime order torsion on elliptic curves over number fields. Part I: Asymptotics." pith.science (2026). https://pith.science/paper/R6PQTNLY

@misc{pith2026250514109,
  author       = {Pith},
  title        = {Pith review of: Prime order torsion on elliptic curves over number fields. Part I: Asymptotics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R6PQTNLY}},
  note         = {Machine review of arXiv:2505.14109}
}
abstract

We study the asymptotics of the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$ as $d$ tends to infinity. Assuming some conjectures on the sparsity of newforms of weight $2$ and prime level with unexpectedly high analytic rank, we show that $\max S(d) \le 3d + 1$ for sufficiently large even $d$ and $\max S(d) = o(d)$ for odd $d$.

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Works this paper leans on

7 extracted references · 6 canonical work pages

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