On the $p$-adic distribution of torsion values for a section of an abelian scheme

Brian Lawrence; Umberto Zannier

On the $p$-adic distribution of torsion values for a section of an abelian scheme

T0 review · 0 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For a section of an abelian scheme over a p-adic field, torsion values of different exact orders are separated by rigid-analytic neighborhoods, and non-torsion points are p-adically isolated from the torsion locus.

desk verdict A clean, correct note: the torsion locus of a section of an abelian scheme separates p-adically by exact torsion order, with a sharpness example that shows the abelian hypothesis is needed. read the letter →

arxiv 1908.09050 v1 pith:5MRYPMLG submitted 2019-08-23 math.NT

Brian Lawrence , Umberto Zannier This is my paper · ORCID

classification math.NT MSC 11G1014G2214K15

keywords p-adicfieldsabelianschemestorsionlocusrigidanalyticspacesformalgroupsTateuniformizationbadreductionunlikelyintersections

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a p-adic separation theorem for torsion values of a section of an abelian scheme. The main theorem says that around any point of the base variety, torsion values have a single exact order: a point that is torsion of order n has a rigid-analytic neighborhood containing no torsion of any other order, and a non-torsion point has a neighborhood with no torsion at all. This gives a precise sense in which torsion points of different orders stay away from each other over p-adic fields, in contrast to the complex-analytic setting where torsion values can be dense. The proof works by constructing, near each point, a rigid-analytic open subgroup of the abelian scheme that is isomorphic to the unit ball and fiberwise of finite index; torsion detection then reduces to a single linear equation. The paper closes with an example showing the conclusion fails if the abelian scheme is allowed to have bad reduction.

What carries the argument

The machinery is a local structure theorem for abelian schemes over p-adic fields. It says that around each K-point of the base there is a rigid-analytic open set E in the total space, containing the identity section, that is isomorphic to U times the g-dimensional unit ball in such a way that the group law is coordinatewise addition and the intersection of E with each fiber is a subgroup; moreover this subgroup has finite index in the fiber over the chosen point. Relative translation-invariant differentials are used to write the group law as addition in power-series coordinates, and p-adic compactness gives the finite-index statement. Once E is available, a single multiple n brings the section into the torsion-free group E throughout a neighborhood, so the torsion condition becomes ns(x)=0 and all nearby torsion has the same exact order.

What would settle it

Look for a smooth fiber at which torsion values of infinitely many exact orders accumulate: for a section of an abelian scheme over a p-adic field, if there were a sequence x_m converging to x_0 with x_0 in the smooth locus and s(x_m) of pairwise distinct exact torsion orders, Theorem 1.2 would fail. The paper's Section 3 example shows the analogous accumulation does occur at a singular fiber, so smoothness is the decisive distinction.

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Extended reading notes

Core claim

Write S_n for the subvariety of the base where the section is torsion of exact order n. The central claim is that for any finite extension L of the p-adic field K, every point x0 in S_n(L) has a rigid-analytic neighborhood U with U disjoint from S_n' for n' different from n, and every point x0 not in any S_n has a rigid-analytic neighborhood disjoint from all S_n. In particular, the torsion locus is a disjoint union of p-adically open pieces indexed by exact torsion order, and no torsion point of one order can be approached by torsion points of other orders. The same conclusion is shown to be false for families with bad reduction: the paper constructs an explicit elliptic scheme over the p-adic disk with a section whose torsion points of every sufficiently large exact order accumulate at the singular fiber.

Load-bearing premise

The premise that carries the argument is that at every point the abelian scheme contains a rigid-analytic open subgroup isomorphic to the unit ball whose intersection with the given fiber has finite index; if this finite-index condition fails, nearby torsion orders could vary and the theorem's conclusion would not follow.

Editorial extensions

If this is right

No sequence of torsion points with pairwise distinct exact orders can converge in S(K): any limit would be torsion of one order or non-torsion, and both cases are excluded by the theorem.
The exact torsion order is locally constant on the torsion locus: each S_n is open in the rigid topology of the union of the S_n's.
A non-torsion point has a rigid-analytic neighborhood free of torsion values, so the torsion locus has empty interior and is not topologically dense in S(K).
The hypothesis that the fibers are abelian varieties is necessary in general: the explicit family in Section 3 has torsion of every large order accumulating at a singular fiber.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One might expect a quantitative version: over a fixed quasicompact rigid subspace with good reduction, the p-adic distance between torsion values of different orders could be bounded below uniformly, with the bound depending on the heights of the orders; the theorem gives the qualitative separation but no such bound.
The same local-group argument would likely work for multisections or finite collections of sections, provided each section satisfies the finite-index condition near a point; this would give a p-adic analogue of simultaneous separation.
The counterexample at bad reduction suggests that torsion orders can concentrate only where the identity component of the special fiber degenerates, so the natural global formulation of this phenomenon may live on the smooth locus of a Néron model.
If the theorem is right, complex-density results for torsion values have no direct p-adic analogue: in the rigid topology the torsion locus is locally confined to at most one exact order per open set.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: Theorem 1.2 is correct and the paper is a clean contribution. The proof is short: Lemma 2.1 gives, around any point, a rigid-analytic open subgroup E of A that is fiberwise isomorphic to the unit ball and has finite index in the fiber at the base point. That finite-index property is the load-bearing step, and it follows from p-adic compactness; once you have ns(x) in E for nearby x, torsion-freeness of E forces exact order n, and Zariski closedness excludes proper divisors. No gap I can find.

What's actually new: Scanlon's theorem is for a fixed abelian variety; this handles a family and gives a uniform local statement about exact torsion orders. The proof via the finite-index open subgroup is a neat adaptation. The paper also gives a sharpness example: when the fiber degenerates to a nodal cubic, torsion points of exact order n accumulate at the singular fiber. The construction via Tate uniformization is explicit and convincing; it shows the abelian hypothesis is needed.

Soft spots: these are minor. The power series computations in Section 3 are partly asserted rather than derived. I trust them, but a referee should ask for one or two intermediate steps. The notation n is overloaded (order of torsion vs. index in the proof of Theorem 1.2); that will trip up a careful reader, but it is not a mathematical issue. The paper admits computations are left to the reader; that is fine for a short note.

I do not see a hidden load-bearing flaw. The circularity concern is nil: Lemma 2.1 is proved from smoothness and compactness, and the counterexample is independent. One might ask whether the main theorem could be derived from Scanlon directly by looking at subvarieties of the universal abelian scheme, but the finite-index subgroup argument is genuinely different and handles exact-order separation in a way that would not follow automatically.

I would send this to a reading group on p-adic unlikely intersections or rigid-analytic geometry. It deserves a serious referee, and I would accept.

Referee Report

0 major / 7 minor

Summary. Let K be a p-adic field, S a quasi-projective K-variety, A→S an abelian scheme, and s:S→A a section. For each n, let S_n be the locus where s is torsion of exact order n. The paper proves (Theorem 1.2) that every L-point of S, for a finite extension L/K, has a rigid-analytic neighborhood on which the exact torsion order is constant if the point is torsion, or on which there is no torsion at all if the point is not torsion. The proof relies on Lemma 2.1, which states that around any point of S there is a rigid-analytic open subgroup E of A, isomorphic over a neighborhood U to U×B^g with additive group law, and of finite index in the fiber over that point. Choosing n so that ns(x0)∈E, the torsion-freeness of E(L) forces any torsion point in U to be killed by n, and Zariski closedness of the proper-order torsion loci excludes smaller orders. The paper ends with an explicit elliptic curve family over a p-adic disk, with a singular fiber at t=0 and a section whose torsion orders n occur at points t_n tending to 0, showing that the abelian scheme hypothesis is essential.

Significance. The paper is a short, well-written note that establishes a clean p-adic separation property for torsion values of sections of abelian schemes. The main theorem is a natural counterpart to complex-density results and is proved by an elegant combination of p-adic compactness, logarithmic coordinates, and Zariski closedness of torsion loci; the argument is self-contained once standard rigid-analytic facts are admitted. The explicit counterexample via Tate uniformization is valuable and, despite some small typographical issues, convincingly shows optimality of the hypotheses. I also checked the potential concern that high p-power torsion points might accumulate at the identity and violate the torsion-freeness of the subgroup E: in the logarithmic coordinates E is a small ball where the group law is additive, and p^n-torsion points have coordinates tending to the boundary of the formal group rather than to zero, so they are excluded by taking E sufficiently small. The result should be of interest to researchers working on p-adic unlikely intersections and p-adic dynamics.

minor comments (7)

[§2, proof of Theorem 1.2] The phrase 'the order of the image of s(x0) in the finite group A_{x0}(K)/((E ∩ A_{x0})(K))' is imprecise because the quotient is a finite set of cosets and need not be a group; the intended meaning is the least positive n with ns(x0) ∈ E, which exists by finiteness.
[§3, Lemma 3.3] In Lemma 3.3 the conclusion should include the constant term a2: the inverse is a2 + rZ_p[[f/r^2, (x1-a1)/r^2]] in general, not rZ_p[[...]]. In the application to Proposition 3.4, a2 = p lies in rZ_p because r has valuation 1, so the printed conclusion is correct there; please state the lemma with the constant term or with the hypothesis a2 ∈ rZ_p.
[§3, Proposition 3.4] The power series ring is stated as pZ_p[[(X-X(0,p))/p^3, q/p^3]], but Lemma 3.3, applied with r = ∂X/∂z(0,p)·p (a unit times p), gives the first variable normalized by p^2 rather than p^3; the convergence domain X ∈ X0 + p^4Z_p, q ∈ p^4Z_p is stricter than necessary. Please correct the exponent or the stated radii.
[§3, proof of Proposition 3.1] The congruence for the solution of φ(t) = ŝ(t)^n, stated as t ∈ p^n + p^{n+1}Z_p, appears off by a factor of p: the displayed leading terms φ(t) = -(p/(1-p)^2)t + O(t^2) and ŝ(t)^n = p^n(1+O(t)) give t ∈ p^{n-1} + p^nZ_p. Since the argument only needs t_n → 0, the conclusion is unaffected.
[§3, equation of E_t] There are sign inconsistencies in the displayed family: with the given equation y^2 = (x-1/12)^2(x+1/6) + t(x - p/(1-p)^2 - 1/12), the coefficient B2 should be 1/864 - t(p/(1-p)^2 + 1/12), not 1/864 - t(p/(1-p)^2 - 1/12); consequently the stated leading term of 1/j(E_t) should be +p/(1-p)^2 t + O(t^2), not -p/(1-p)^2 t + O(t^2). These signs should be corrected consistently.
[§2, proof of Theorem 1.2(2)] In part (2), when taking U = U0 - S_n, it would be helpful to state explicitly that S_n ∩ U0 is closed in U0 (it is the zero locus of ns after the proper-order loci are excluded), so that the complement is a rigid-analytic open neighborhood.
[References] Reference [1] appears to be a preprint; please add the arXiv identifier or publication data.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found; the proof of Theorem 1.2 is self-contained, with Lemma 2.1 proved from smoothness and p-adic compactness, and the counterexample constructed independently.

full rationale

The paper derives Theorem 1.2 from Lemma 2.1, and Lemma 2.1 is proved inside the paper rather than imported from the authors' prior work. Properties (1) and (3) follow from smoothness of the abelian scheme and explicit integration of translation-invariant differentials, while property (4) follows from compactness of A_{x0}(K) together with the fact that cosets of the open subgroup are open. There is no fitted parameter, no definition of a quantity in terms of the result being proved, and no invocation of a self-citation as the load-bearing step. The references to prior work in the introduction are contextual and motivational, not used to justify the theorem. The Section 3 counterexample is an independent explicit construction showing that the conclusion genuinely fails when the abelian scheme hypothesis is dropped; it does not assume the main theorem. The proof of the counterexample uses standard Tate uniformization, explicit power-series computations, and Lemma 3.5, all proved or stated with sufficient detail, so there is no renaming of a known result as an organizing principle. Thus the paper is self-contained against external benchmarks and no circular step is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new constants, particles, forces, or ad hoc objects. The counterexample's power series are explicit and fit into the standard Tate uniformization framework.

assumptions (4)

domain assumption Rigid analytic geometry over a p-adic field is a valid analytic framework for finite-type K-schemes; the analytic topology and affinoid theory from [3] are used without reproof.
Invoked throughout: the analytification functor, affinoids, and the open subspaces defined by |f| <= 1. Standard for p-adic arithmetic geometry.
standard math An abelian scheme over a characteristic-zero base is smooth, and its sheaf of translation-invariant differentials is locally free; these differentials can be locally integrated as formal power series.
Used in the proof of Lemma 2.1 to construct the analytic coordinates u_i and to show the group law is additive on the neighborhood E.
standard math The p-adic inverse and implicit function theorems and the binomial theorem for power series over Z_p hold as stated.
Used repeatedly: to get analytic isomorphisms in Lemma 2.1, to invert X(q,z) in Proposition 3.4, to take square roots in Lemma 3.5, and to solve phi(t)=s-hat(t)^n by successive approximation.
domain assumption Tate uniformization describes elliptic curves over a p-adic field with split multiplicative reduction as G_m/q^Z, with the given X(q,z) and Y(q,z) series and j-invariant expansion.
The counterexample in Section 3 builds the family E_t from the Tate model and uses the expression for 1/j(E_t) and the leading coefficients of a4 and a6; these are standard from [11] up to the stated affine transformation.

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Pith. "Pith review of On the $p$-adic distribution of torsion values for a section of an abelian scheme." pith.science (2026). https://pith.science/paper/5MRYPMLG

@misc{pith2026190809050,
  author       = {Pith},
  title        = {Pith review of: On the $p$-adic distribution of torsion values for a section of an abelian scheme},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5MRYPMLG}},
  note         = {Machine review of arXiv:1908.09050}
}

abstract

Let $A \rightarrow S$ be an abelian scheme over a $p$-adic field, and let $s \colon S \rightarrow A$ be a section. We study the torsion locus $\bigcup \limits_{n \geq 1} s^{-1}(A[n])$ on $S$, and we show that torsion points on $S$ of different orders stay away from each other.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On the torsion values for sections of an elliptic scheme
math.AG 2019-09 conditional novelty 7.0 of 10

The canonical height of a section of an elliptic scheme over a curve equals the integral of the Betti form over the base, and this measure coincides with the dynamical equidistribution measure.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · cited by 1 Pith paper

[1]

André, P

Y. André, P. Corvaja, U. Zannier, with an appendix by Z. Ga o. The Betti map associated to a section of an abelian scheme

work page
[2]

Bogatyrev

A. Bogatyrev. Eﬀective computation of Chebyshev polyno mials for several intervals. Sb.Math, 190:11 (1999), pp. 1571–1605

work page 1999
[3]

Bosch, U

S. Bosch, U. Güntzer, R. Remmert. Non-Archimedean Analysis. Springer-Verlag, 1984

work page 1984
[4]

Corvaja, D

P. Corvaja, D. Masser, and U. Zannier. Torsion hypersurfaces on abelian schemes and Betti coordin ates. Math. Ann. 371:3–4 (2018), 1013–1045

work page 2018
[5]

DeMarco and N

L. DeMarco and N. Mavraki, Variation of Canonical Height and Equidistribution , preprint 2017

work page 2017
[6]

Lawrence

B. Lawrence. A density result for real hyperelliptic curves . Comptes Rendus Mathematique, 354:12 (2016), 1219–1224

work page 2016
[7]

Maulik and B

D. Maulik and B. Poonen. Néron–Severi groups under specialization . Duke Math. J., 161:11 (2012), 2167– 2206

work page 2012
[8]

Masser and U

D. Masser and U. Zannier. Torsion points, Pell’s equations and integration in ﬁnite t erms, preprint 2018

work page 2018

Show all 13 references

[9]

T. Scanlon. The conjecture of Tate and Voloch on p-adic proximity to torsion . Internat. Math. Res. Notices 17 (1999), 909–914

1999
[10]

J-P. Serre. Distribution asymptotique des valeurs propres des endomor phismes de Frobenius [d’après Abel, Chebyshev, Robinson,...], Séminaire BOURBAKI 1146 (2017– 2018)

2017
[11]

Silverman

Joseph H. Silverman. Advanced Topics in the Arithmetic of Elliptic Curves . Springer-Verlag, 1994. 8 BRIAN LA WRENCE AND UMBERTO ZANNIER

1994
[12]

C. Voisin. Torsion points of sections of Lagrangian torus ﬁbrations an d the Chow ring of hyper-Kähler fourfolds, preprint (2017), ArXiv:1603.04320v3[mathAG]7Jan2018

2017 arXiv
[13]

U. Zannier. Unlikely Intersections and Pell’s Equations in polynomial s. Springer INdAM Series 8, V. Ancona and E. Strickland Eds., 2014

2014

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

André, P

Y. André, P. Corvaja, U. Zannier, with an appendix by Z. Ga o. The Betti map associated to a section of an abelian scheme

work page

[2] [2]

Bogatyrev

A. Bogatyrev. Eﬀective computation of Chebyshev polyno mials for several intervals. Sb.Math, 190:11 (1999), pp. 1571–1605

work page 1999

[3] [3]

Bosch, U

S. Bosch, U. Güntzer, R. Remmert. Non-Archimedean Analysis. Springer-Verlag, 1984

work page 1984

[4] [4]

Corvaja, D

P. Corvaja, D. Masser, and U. Zannier. Torsion hypersurfaces on abelian schemes and Betti coordin ates. Math. Ann. 371:3–4 (2018), 1013–1045

work page 2018

[5] [5]

DeMarco and N

L. DeMarco and N. Mavraki, Variation of Canonical Height and Equidistribution , preprint 2017

work page 2017

[6] [6]

Lawrence

B. Lawrence. A density result for real hyperelliptic curves . Comptes Rendus Mathematique, 354:12 (2016), 1219–1224

work page 2016

[7] [7]

Maulik and B

D. Maulik and B. Poonen. Néron–Severi groups under specialization . Duke Math. J., 161:11 (2012), 2167– 2206

work page 2012

[8] [8]

Masser and U

D. Masser and U. Zannier. Torsion points, Pell’s equations and integration in ﬁnite t erms, preprint 2018

work page 2018