Explicit bounds on common projective torsion points of elliptic curves

Christian B\"ohning; David Hubbard; Hans-Christian Graf von Bothmer

REVIEW 3 major objections 6 minor 43 references

Explicit bounds on common projective torsion points of elliptic curves

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read When two elliptic curves with standard double covers have good reduction at a prime $p$, the paper bounds their common torsion images by $2p^3+8$, with refinements down to $2p+8$; in bad multiplicative reduction it gives the conditional…

desk verdict The good-reduction bound is a real effective result, but the bad-reduction theorem is conditional on an unproved finiteness assumption that the abstract understates. read the letter →

arxiv 2412.20174 v1 pith:Z4BJZ3MX submitted 2024-12-28 math.AG

Christian Böhning , Hans-Christian Graf von Bothmer , David Hubbard This is my paper · ORCID

classification math.AG MSC 11G0514H5214K1214G20

keywords commonprojectivetorsionpointsellipticcurvesdoublecoversunlikelyintersectionseffectiveboundslogarithmicalgebraicgeometryWittvectorson

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

This paper asks a quantitative version of a uniformity question in unlikely intersections: for two elliptic curves with the standard double cover that identifies a point with its inverse, how many torsion points of one curve can have the same image in the projective line as torsion points of the other? Earlier work proved that the number is bounded independently of the curves, but did not provide realistic constants. The main new result is an explicit bound: if both curves have good reduction at a prime $p$ and the branch loci of the two covers are disjoint on the special fibre, then the number of common images of torsion points of order coprime to $p$ is at most $2p^3+8$. Refinements improve this to $2p^2+8$ or $2p+8$ depending on whether the reductions are supersingular or ordinary and whether certain $p$-torsion group schemes split. For bad multiplicative reduction the paper gives the analogous bound $2p^3+2$, under a finiteness assumption that it verifies only in special cases; these are the first effective bounds in reach of numerical testing.

What carries the argument

The load-bearing construction is the first-order infinitesimal deformation bundle. One works with the abelian scheme $A=E_1\times E_2$ over the Witt vectors $R=W(\mathbb{F}_p)$ and with $X=(\pi_1\times\pi_2)^{-1}(\Delta)$, the preimage of the diagonal in $\mathbb{P}^1\times\mathbb{P}^1$. Torsion of order coprime to $p$ specialises injectively into the central fibre, and the paper bounds its image by $|\mathrm{im}(pA_1(R_1)\cap X_1(R_1)\to X_0(k))|$, where $R_1=R/p^2$. Points of $X_1(R_1)$ lifting a point of $X_0$ determine normal directions, encoded in the affine bundle $V_0=\mathbb{P}(N_{X_0/A})\setminus\mathbb{P}(N_{X_0/A_0})$; the image of $X_1(R_1)$ is a curve $X'_0$ over $X_0$, and multiplication by $p$ on $A_0$, which factors through the relative Frobenius morphism, produces a curve $Y'_0$ that is numerically $\delta X'_0$. Intersection theory gives $X'_0\cdot Y'_0=8\delta$, and the Frobenius factorisation gives $\delta\leq p^3$. In bad multiplicative reduction the same counting problem is moved to $\mathbb{P}^1\times\mathbb{P}^1$ through the rational multiplication-by-$p$ maps, and a bidegree computation with lifted polynomials (Proposition 6.3) supplies the $p^3$ bound.

What would settle it

A direct calculation for an explicit pair satisfying the good-reduction assumptions at a small prime $p$ would settle the main bound: the theorem predicts at most $2p^3+8$ common projective torsion points of order coprime to $p$, so a pair with more would refute it. For the bad-reduction theorem, a concrete bad-multiplicative-reduction model with a $p$-th power uniformiser and a good ordinary partner for which the first-order lift set reduces to infinitely many points on the special fibre would disprove the finiteness claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 2.5: under Assumptions 1.2 and 2.4, $t(E_1,\pi_1,E_2,\pi_2,p') \leq 2p^3+8$, where $t$ counts points of the projective line that are images, under both standard double covers, of torsion points of order not divisible by $p$. The proof routes the count through torsion points of the abelian surface $A=E_1\times E_2$ lying on the curve $X=(\pi_1\times\pi_2)^{-1}(\Delta)$, the preimage of the diagonal, and then through first-order deformations over the Witt vectors; the geometry of an affine bundle over the special fibre reduces the problem to bounding an intersection number by $8\delta$ with $\delta\leq p^3$. Refinements in Section 3 sharpen $\delta$ to $p^2$ or $p$ in the supersingular and split-ordinary cases. In bad multiplicative reduction, Theorem 5.9 transfers the argument to a product of projective lines and obtains $|M|\leq 2p^3+2$ conditional on Assumption 5.2(b), and Theorem 7.1 proves that finiteness when the first curve's bad-reduction model has a $p$-th power uniformiser and the second has good ordinary reduction.

Load-bearing premise

The load-bearing premise is Assumption 5.2(b), which says that the set of first-order lift points lying on the common-torsion curve after multiplication by $p$ is finite; the paper proves this only under extra hypotheses and leaves the general bad-reduction case open.

Editorial extensions

If this is right

For any pair of elliptic curves with standard double covers satisfying the good-reduction assumptions at a prime $p$, the number of common projective torsion images of order coprime to $p$ is at most $2p^3+8$, a constant small enough to be compared with computational torsion data.
When both special fibres are supersingular, the bound improves to $2p^2+8$; when both are ordinary and the relevant connected-étale sequences split, it improves to $2p+8$.
Combining the $p$-prime bound with a second prime $q$ and a large-Galois-orbit condition yields a finite total bound $c=8p^{4r+3}$ on all torsion points of $A=E_1\times E_2$ lying on $X$.
In the bad multiplicative reduction case, under Assumption 5.2 the number of coprime-to-$p$ torsion pairs with a common projection is at most $2p^3+2$.
The finiteness needed for the bad-reduction bound is proved unconditionally when the first curve's model has a $p$-th power uniformiser and the second curve has good ordinary reduction.

Reading between the lines

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

A natural testable extension, not carried out in the paper, is to run the algorithmic canonical-lift criterion of Section 3 on explicit curves for small primes and compare the refined bounds $2p^2+8$ and $2p+8$ with brute-force torsion computations.
If the finiteness in Assumption 5.2(b) is proved in general, the bad-multiplicative-reduction theorem becomes unconditional, and the restriction to a $p$-th-power uniformiser is likely removable.
The affine-bundle intersection technique suggests a route to higher-dimensional abelian varieties or higher-genus curves, with the degree of multiplication-by-$p$ on the relevant subvariety replacing the role of $p^3$.
The explicit constant opens up a computational search for the true optimal value: enumerating torsion images for many pairs at a fixed small $p$ could show whether $2p^3+8$ is close to the maximum.

Share X Bluesky LinkedIn Reddit HN

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

You should know two things about this paper. First, the good-reduction part is a genuine step forward: Theorem 2.5 gives t <= 2p^3+8 for common images of torsion coprime to p, and the refinements for supersingular and ordinary cases (2p^2+8 and 2p+8) are natural and plausible. Second, the bad-reduction Theorem 5.9 is conditional on Assumption 5.2(b), a finiteness statement that is close to the target conclusion; the authors prove it only under special hypotheses (Theorem 7.1) and admit in Remark 5.3 that the general implication is open. The abstract's phrase "mild extra assumptions" does not convey that the multiplicative-reduction case rests on an unproved hypothesis.

What is actually new: the extension of Raynaud's method to bad multiplicative reduction using log algebraic geometry, the admissible factorization of multiplication-by-p in Proposition 7.4, and the bound in Proposition 6.3 on sections of rational maps with zero differential. The paper builds on Raynaud's classical intersection-counting framework rather than the ineffective uniformity results, so it is a substantial technical extension rather than a new paradigm. The degree estimate for the reduced preimage under multiplication by p (the p^3 bound) is a bit quick but plausible; the overall proof of Theorem 2.5 is coherent.

The soft spots, in proportion: (1) Assumption 5.2(b) is the load-bearing issue in the bad reduction case. If it fails, the bound |M| <= 2p^3+2 collapses. The paper is honest about the gap, but the packaging is not. (2) The full torsion bound in Section 4 involves a constant c = 8p^{4r+3} where r depends on the Galois action, so it is explicit in principle but not "realistic" in the sense advertised. (3) Lemma 4.3 has a terse step for a not rational; the Galois-conjugate argument is standard but the intersection bound could use a line of detail. These are minor relative to Assumption 5.2(b).

The citation pattern is fine; there is no dependence on the authors' own prior results, and no data fitting or fabrication concerns. The paper deserves a serious referee. I would send it to a good journal, but the referee should push for either a proof of Assumption 5.2(b) in greater generality or a clear separation of unconditional and conditional results, and the abstract should be rewritten to state the finiteness hypothesis explicitly. This is not a desk reject.

Referee Report

3 major / 6 minor

Summary. The paper studies the Bogomolov-Fu-Tschinkel problem on the number of common images of torsion points of two elliptic curves under standard double covers to the projective line. It proves explicit bounds on the number of common images of torsion points of order coprime to a prime p under a good-reduction assumption (Theorem 2.5: t ≤ 2p^3+8), with refinements in the supersingular and ordinary cases (Propositions 3.2 and 3.3). It also states a bound in the case of bad multiplicative reduction (Theorem 5.9: |M| ≤ 2p^3+2) conditional on a finiteness assumption (Assumption 5.2(b)), and develops logarithmic-geometric techniques (Sections 7–10) to verify this assumption in a special case (Theorem 7.1). Section 4 attempts to combine bounds at two primes to control the full torsion set.

Significance. If the good-reduction bound of Theorem 2.5 is correct, it provides the first explicit, effective bound for the prime-to-p part of the common projective torsion problem, complementing the ineffective uniformity results cited in the introduction. The adaptation of Raynaud's Manin-Mumford method and the log-geometric machinery for bad reduction are original and constitute a substantive technical contribution. The main qualifications are that the bad-reduction theorem is conditional on an unproved finiteness statement and that the proposed combination step for full torsion (Section 4) is not fully justified. As a result, the paper's core unconditional contribution is the prime-to-p bound in the good-reduction case, not a complete solution of the original problem for all torsion points.

major comments (3)

[Section 5, Assumption 5.2(b), Theorem 5.9] The bound |M|≤2p^3+2 in Theorem 5.9 is conditional on the finiteness of im(pA_1^∘(R_1)∩X_1^∘(R_1)→X_0^∘(k)) stated in Assumption 5.2(b). This assumption is not proved in general; as Remark 5.3 explicitly concedes, the authors only establish it under the restrictive hypotheses of Theorem 7.1 (E_1 with Tate parameter q=π^p and E_2 with good ordinary reduction). Since the proof of Theorem 5.9 injects M into exactly this image, the multiplicative-reduction case is not established unconditionally, and the abstract's phrasing "mild extra assumptions" overstates what is proved.
[Section 4, Lemmas 4.3 and 4.4] The deduction of Lemma 4.4 (t_{A,X+a,p}≤8q^3) from Lemma 4.3 by "repeating the above argument for a second prime q" is not justified. The proof of Lemma 4.3 for a∉A(K) uses the Galois group of K, where K is the completion of the maximal unramified extension at p, and the unramifiedness of A[∞](p') at p. To obtain the analogous statement at q one would need to pass to a different base field (in general K(a)), show that the translated curve X+a has a suitable good-reduction model at q, and re-run the argument with q-primary torsion; none of these steps is addressed. Since Proposition 4.2 relies on Lemma 4.4 to combine the p'-torsion and p-primary bounds, the claimed control of the full torsion set is not proved.
[Section 2, proof of Theorem 2.5] The estimate δ≤p^3 for the degree of the map Y'_0→X_0 is compressed into the sentence "Since Y_0 is defined to be the reduced preimage of X_0 under this map, we get the desired bound." The argument should explicitly exhibit the factorization of [p]|Y_0 through the relative Frobenius and bound the degrees of the two factors, taking care that Y_0 is reduced and may have components on which the degrees differ. As written, this step is too quick for a load-bearing bound, although it is likely fixable by spelling out the standard inseparability-degree argument.

minor comments (6)

[Section 8, Proposition 8.1] The sentence "The curve Γ is a connected curve with N connected components each of which is a nonsingular curve of genus 3" is contradictory in its current wording; "connected components" should presumably be "irreducible components", and the following sentence about intersections in nodes should be made consistent with that reading.
[Sections 2 and 5] The main theorems do not state that p is odd, yet Section 8 explicitly assumes p≠2 for the arguments leading to Theorem 7.1. The paper should state clearly whether Theorems 2.5 and 5.9 are intended also for p=2, and if so, indicate what modifications are needed.
[Introduction and abstract] The abstract and introduction promise bounds for common projective torsion points under "mild extra assumptions", but the bad-reduction theorem is conditional on Assumption 5.2(b). The authors should explicitly qualify this in the abstract and introduction so that the conditional nature of the multiplicative-reduction results is clear to the reader.
[Section 4, first paragraph] There is a duplicated phrase: "we have we have only discussed" should read "we have only discussed".
[Section 10, Lemma 10.2] The statement of Lemma 10.2 contains the typo "Suppose that the the assumptions", which should be corrected.
[Section 10, notation] The notation "where we denote a uniformiser of R by π" in the log smoothness discussion conflicts with the use of π for the projections π_i, π_{i,R}; using a different symbol such as ϖ for the uniformiser would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the good-reduction bound is derived from Raynaud-type intersection calculations, and the bad-reduction theorem is conditional on an openly stated finiteness hypothesis rather than a circular reduction.

full rationale

The paper's central derivation is self-contained against its stated assumptions. In Section 2, Proposition 2.3 reduces the coprime-to-p torsion count to |im(pA1(R1)∩X1(R1)→X0(k))|, and Theorem 2.5 bounds this by the intersection number X'_0·Y'_0 = 8δ, with δ controlled by the degree of multiplication by p on the abelian surface. The constants 8 and p^3 come from X0·X0 = 8 and the degree of [p] factoring through Frobenius, not from the quantity being bounded. Section 4's total-torsion bound combines this with Galois-orbit estimates from Serre, Kraus, and Smith, which are external facts. The only structurally delicate point is the bad-multiplicative-reduction case. Theorem 5.9 is explicitly conditional on Assumption 5.2(b), the finiteness of im(pA°_1(R1)∩X°_1(R1)→X°_0(k)). The paper states that assumption verbatim, and Remark 5.3 concedes that the implication from Assumption 5.2(a) to 5.2(b) is not proved in general; Theorem 7.1 proves it only under additional hypotheses (Tate parameter q = π^p for E1 and good ordinary reduction for E2). This is a genuine limitation on the advertised bad-reduction scope and belongs under correctness/completeness risk. It is not a circularity in the sense required here: Assumption 5.2(b) is not the same as the claimed bound |M| ≤ 2p^3 + 2, and the proof from the assumption to the bound runs through the independent degree estimate of Proposition 6.3. No fitted parameter is renamed as a prediction, and no load-bearing claim is justified by the present authors' own prior work; the Raynaud citations are external foundational results. Thus no circular step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no hidden fitted constants and no new physical or mathematical entities. Its main external inputs are Raynaud's theorems, Serre-Tate theory, log deformation theory, and standard facts about Neron models. The only substantial ad hoc input is Assumption 5.2(b), a finiteness hypothesis that is the main unresolved point in the bad reduction part.

assumptions (5)

standard math Raynaud's Proposition 3.3.1 and Theorem 4.4.1 describing the affine bundle V0 and intersection numbers of X'_0 and Y'_0.
Used in the proof of Theorem 2.5 to pass from R1-valued points to intersections of curves in a projective bundle over X0.
ad hoc to paper Assumption 5.2(b): finiteness of im(pA_1^o(R1) cap X_1^o(R1) -> X_0^o(k)).
Needed for Theorem 5.9; not proved in general, only under the extra hypotheses of Theorem 7.1.
standard math Serre-Tate canonical lift theory and splitting criteria for connected-etale sequences over W2(k).
Used in Section 3 and Proposition 4.6 to describe Galois orbits and to refine bounds for ordinary and supersingular reductions.
standard math Log smooth deformation theory and non-liftability of relative Frobenius for log smooth curves, as in Kato, Gross and Raynaud's Lemma I.5.4.
Used in Section 10 to prove Theorem 7.1, the special case where Assumption 5.2(a) implies Assumption 5.2(b).
domain assumption The spread construction in Remark 1.1 reduces curves over C to number fields while preserving the finiteness of torsion intersections.
Bridges from complex elliptic curves to models over number fields and assumes that the number of common branch points is constant in the family.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Explicit bounds on common projective torsion points of elliptic curves." pith.science (2026). https://pith.science/paper/Z4BJZ3MX

@misc{pith2026241220174,
  author       = {Pith},
  title        = {Pith review of: Explicit bounds on common projective torsion points of elliptic curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z4BJZ3MX}},
  note         = {Machine review of arXiv:2412.20174}
}

read the original abstract

Suppose E_1, E_2 are elliptic curves (over the complex numbers) together with standard double coverings of the projective line identifying a point and its inverse on E_i. Bogomolov, Fu and Tschinkel have asked if the number of common images of torsion points on the elliptic curves under these double coverings is uniformly bounded in the case when the branch loci of the double coverings do not coincide, and recently this was answered affirmatively by various authors, but realistic effective bounds are unknown. In this article we obtain such bounds for common projective torsion points on elliptic curves under some mild extra assumptions on the reduction type of the input data at given primes. The method is based on Raynaud's original groundbreaking work on the Manin-Mumford conjecture. In particular, we generalise several of his results to cases of bad reduction using techniques from logarithmic algebraic geometry.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 39 canonical work pages

[1]

Bogomolov, F., Fu, H., Division polynomials and intersection of projective torsion points, Eur. J. Math. 2 (2016), no. 3, 644--660

work page 2016
[2]

Bogomolov, F., Fu, H., Elliptic curves with large intersection of projective torsion points, Eur. J. Math. 4, 555--560 (2018)

work page 2018
[3]

Bogomolov, F., Fu, H., Tschinkel, Y., Torsion of elliptic curves and unlikely intersections, Geometry and physics. Vol. I, Oxford Univ. Press, Oxford, (2018), 19--37

work page 2018
[4]

Coleman, R.F., Torsion points on curves, Advanced Studies in Pure Mathematics 12, in: Galois Representations and Arithmetic Algebraic Geometry (1987), 235--247

work page 1987
[5]

Deligne, P., Rapoport, M., Les sch\' e mas de modules de courbes elliptiques , in: ``Modular Functions of One Variable II", W. Kuyk, P. Deligne (eds.), Lecture Notes in Math. 349, Springer (1973)

work page 1973
[6]

DeMarco, L., Krieger, H., Ye, H., Uniform Manin-Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), no. 3, 949--1001

work page 2020
[7]

Dimitrov, V., Gao, Z., Habegger, P., Uniformity in Mordell-Lang for curves, Annals of Math., 194 (1), 237--298

work page
[8]

Edixhoven, B., van der Geer, G., Moonen, B., Abelian Varieties, online book preprint, available at http://van-der-geer.nl/ gerard/AV.pdf http://van-der-geer.nl/ gerard/AV.pdf

work page

Show all 43 references

[9]

Esnault, H., Viehweg, E., Lectures on Vanishing Theorems, DMV Seminar Band 20, Springer Basel AG (1992)

1992
[10]

Finotti, L., Lifting the j-invariant: Questions of Mazur and Tate, Journal of Number Theory 130, number 3, pages 620--638, 2010, Elsevier

2010
[11]

Fu, H., Stoll, M., Elliptic curves with common torsion x-coordinates and hyperelliptic torsion packets, Proc. Amer. Math. Soc. 150 (2022), no. 12, 5137--5149

2022
[12]

Gao, Z., Recent developments of the uniform Mordell-Lang conjecture, (2021) https: //arxiv.org/abs/2104.03431v4 https: //arxiv.org/abs/2104.03431v4

2021 arXiv
[13]

Gao, Z., Ge, T., K\"uhne, L., The uniform Mordell-Lang conjecture, (2021), https://arxiv.org/abs/2105.15085v2 https://arxiv.org/abs/2105.15085v2

2021
[14]

Grothendieck (Ed.), Groupes de Monodromie en G\' e ometrie Alg\' e brique, SGA7 I, Lecture Notes in Math., vol

Grothendieck, A., Mod\` e les de N\' e ron et monodromie , in: A. Grothendieck (Ed.), Groupes de Monodromie en G\' e ometrie Alg\' e brique, SGA7 I, Lecture Notes in Math., vol. 288, Springer (Berlin) (1972), 313--523

1972
[15]

Grothendieck, A., Dieudonn\' e , J., \' E l\' e ments de g\' e ometrie alg\' e brique IV , Publ. Math. IHES (1966)

1966
[16]

Gross, M., Tropical Geometry and Mirror Symmetry, CBMS Regional Conference Series in Mathematics, Number 114, AMS (2011)

2011
[17]

257, Springer-Verlag (2010)

Hartshorne, R., Deformation Theory, Graduate Texts in Math. 257, Springer-Verlag (2010)

2010
[18]

Hida, H., Geometric Modular Forms and Elliptic Curves, Second Edition, World Scientific (2012)

2012
[19]

Kato, F., Log smooth deformation theory, T\^ o hoku Math. J. 48 (1996), 317--354

1996
[20]

868 (1978), 138--202

Katz, N.M., Serre-Tate local moduli, in: ``SurfacesAlg\' e briques", Lecture Notes in Math. 868 (1978), 138--202

1978
[21]

Kraus, Alain, D \'e termination du poids et du conducteur associ \'e s aux repr \'e sentations des points de p-torsion d'une courbe elliptique , Polska Akademia Nauk, Instytut Matematyczny
[22]

K\"uhne, L., Equidistribution in families of abelian varieties and uniformity, (2021), https://arxiv.org/abs/2101.10272v2 https://arxiv.org/abs/2101.10272v2

2021 arXiv
[23]

Liu, Q., Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics 6, Oxford University Press (2002), reprint 2010

2002
[24]

Mehta, V.B., Srinivas, V., Varieties in positive characteristic with trivial tangent bundle, Compositio Mathematica, tome 64, no 2 (1987), 191--212

1987
[25]

Milne, J.S., Algebraic groups : the theory of group schemes of finite type over a field, Cambridge studies in advanced mathematics 170, Cambridge University Press (2017)

2017
[26]

Ogus, A., Lectures on Logarithmic Algebraic Geometry, Cambridge studies in adv. math. 178, Cambridge University Press (2018)

2018
[27]

I : Mesures d'\' e quilibre sur une droite projective relative , (2022) https://arxiv.org/abs/2201.08480 https://arxiv.org/abs/2201.08480

Poineau, J., Dynamique analytique sur Z . I : Mesures d'\' e quilibre sur une droite projective relative , (2022) https://arxiv.org/abs/2201.08480 https://arxiv.org/abs/2201.08480

2022 arXiv
[28]

II : \' E cart uniforme entre Latt\` e s et conjecture de Bogomolov-Fu-Tschinkel , (2022) https://arxiv.org/abs/2207.01574 https://arxiv.org/abs/2207.01574

Poineau, J., Dynamique analytique sur Z . II : \' E cart uniforme entre Latt\` e s et conjecture de Bogomolov-Fu-Tschinkel , (2022) https://arxiv.org/abs/2207.01574 https://arxiv.org/abs/2207.01574

2022 arXiv
[29]

Raynaud, M., Passage au quotient par une relation d'equivalence plate, in: Proceedings of a conference on Local Fields, Driebergen, 1966, Springer, (1967), 79--85

1967
[30]

Raynaud, M., Courbes sur une vari\' e t\' e ab\' e lienne et points de torsion , Invent. math. 71, (1983), 207--233

1983
[31]

(eds.), Lecture Notes in Mathematics 1016 Springer (Berlin, Heidelberg) (1983)

Raynaud, M., Around the Mordell conjecture for function fields and a conjecture of Serge Lang, in: ``Algebraic Geometry", Raynaud, M., Shioda, T. (eds.), Lecture Notes in Mathematics 1016 Springer (Berlin, Heidelberg) (1983)

1983
[32]

Monographs, Vol

Saito, T., Fermat's Last Theorem, Basic Tools, Translations of Math. Monographs, Vol. 243, AMS (2013)

2013
[33]

Wissenschaften 334, Springer (2006)

Sernesi, E., Deformations of algebraic schemes, Grundlehren der math. Wissenschaften 334, Springer (2006)

2006
[34]

Serre, J-P., Propri\' e t\' e s galoisiennes des points d'ordre fini des courbes elliptiques , Inv. math. 15, (1972), 259--331

1972
[35]

67, Springer (1979)

Serre, J-P., Local Fields, Graduate Texts in Math. 67, Springer (1979)

1979
[36]

Serre, J-P., Abelian l -adic Representations and Elliptic Curves, Research Notes in Mathematics, Volume 7, A K Peters, Ltd. (1998)

1998
[37]

106, 2nd Edition, Springer-Verlag (2009)

Silverman, J.H., The Arithmetic of Elliptic Curves, Graduate Texts in Math. 106, 2nd Edition, Springer-Verlag (2009)

2009
[38]

151, 2nd Edition, Springer-Verlag (1994)

Silverman, J.H., Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Math. 151, 2nd Edition, Springer-Verlag (1994)

1994
[39]

volume 9, page 17, Springer, 2023

Smith, Hanson, Ramification in division fields and sporadic points on modular curves, Research in Number Theory. volume 9, page 17, Springer, 2023

2023
[40]

Talpo, M., Vistoli, A., Deformation theory from the point of view of fibered categories, (2010), arXiv:1006.0497, 2010

2010 arXiv
[41]

Cornell, J.H

Tate, J., Finite flat group schemes, in: ``Modular Forms and Fermat's Last Theorem", G. Cornell, J.H. Silverman, G. Stevens eds., Springer (1997)

1997
[42]

66, Springer-Verlag (1979)

Waterhouse, W.C., Introduction to Affine Group Schemes, Graduate Texts in Math. 66, Springer-Verlag (1979)

1979
[43]

Studies 181, Princeton University Press (2012)

Zannier, U., Some Problems of Unlikely Intersections in Arithmetic and Geometry, Annals of Math. Studies 181, Princeton University Press (2012)

2012

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Bogomolov, F., Fu, H., Division polynomials and intersection of projective torsion points, Eur. J. Math. 2 (2016), no. 3, 644--660

work page 2016

[2] [2]

Bogomolov, F., Fu, H., Elliptic curves with large intersection of projective torsion points, Eur. J. Math. 4, 555--560 (2018)

work page 2018

[3] [3]

Bogomolov, F., Fu, H., Tschinkel, Y., Torsion of elliptic curves and unlikely intersections, Geometry and physics. Vol. I, Oxford Univ. Press, Oxford, (2018), 19--37

work page 2018

[4] [4]

Coleman, R.F., Torsion points on curves, Advanced Studies in Pure Mathematics 12, in: Galois Representations and Arithmetic Algebraic Geometry (1987), 235--247

work page 1987

[5] [5]

Deligne, P., Rapoport, M., Les sch\' e mas de modules de courbes elliptiques , in: ``Modular Functions of One Variable II", W. Kuyk, P. Deligne (eds.), Lecture Notes in Math. 349, Springer (1973)

work page 1973

[6] [6]

DeMarco, L., Krieger, H., Ye, H., Uniform Manin-Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), no. 3, 949--1001

work page 2020

[7] [7]

Dimitrov, V., Gao, Z., Habegger, P., Uniformity in Mordell-Lang for curves, Annals of Math., 194 (1), 237--298

work page

[8] [8]

Edixhoven, B., van der Geer, G., Moonen, B., Abelian Varieties, online book preprint, available at http://van-der-geer.nl/ gerard/AV.pdf http://van-der-geer.nl/ gerard/AV.pdf

work page