On abelian points of varieties intersecting subgroups in a torus

Jorge Mello

arxiv: 2006.06230 · v6 · pith:JQ4XGC5Unew · submitted 2020-06-11 · 🧮 math.NT · math.AG

On abelian points of varieties intersecting subgroups in a torus

Jorge Mello This is my paper

Pith reviewed 2026-05-24 14:16 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords abelian pointsalgebraic toriconnected algebraic subgroupsnon-anomalous subsetfinitenesssubvarietiesarithmetic dynamics

0 comments

The pith

Under natural conditions the abelian points on the non-anomalous subset of an irreducible subvariety X of a torus, intersected with the union of connected algebraic subgroups of codimension at least dim X, form a finite set.

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

The paper sets out to show that, when certain natural conditions hold, the abelian points lying in the non-anomalous part of the intersection between a closed irreducible subvariety X of a torus and the union of connected algebraic subgroups of codimension at least the dimension of X are finite. This directly generalizes a finiteness statement proved by Ostafe, Sha, Shparlinski and Zannier in 2017. The same work extends the earlier structure theorem to the case in which the subgroups need not be connected and derives an additional statement for curves in the setting of arithmetic dynamics. A sympathetic reader would care because the result limits the number of arithmetically special points that can appear on such intersections inside multiplicative tori.

Core claim

The paper claims that the set of abelian points on the non-anomalous subset of a closed irreducible subvariety X intersected with the union of connected algebraic subgroups of codimension at least dim X in a torus is finite, under natural conditions. It also generalises the structure theorem for such sets when the algebraic subgroups are not necessarily connected and obtains a related result in the context of curves and arithmetic dynamics.

What carries the argument

The non-anomalous subset of the intersection of X with the union of the subgroups, which excludes degenerate cases so that the finiteness statement applies.

If this is right

The finiteness conclusion applies to the non-anomalous part of the stated intersections whenever the natural conditions hold.
The structure theorem for the sets of such points extends from connected subgroups to the non-connected case.
A related finiteness or structure statement holds for curves in the arithmetic-dynamics setting.

Where Pith is reading between the lines

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

If the natural conditions turn out to be satisfied for large classes of varieties, the result would restrict the arithmetic points that can appear on many subvarieties of tori.
The codimension condition on the subgroups may serve as a template for obtaining similar control in other ambient algebraic groups.

Load-bearing premise

The natural conditions on the variety, the subgroups, and the definition of the non-anomalous subset are satisfied so that degenerate intersections are excluded.

What would settle it

An explicit irreducible subvariety X together with a family of connected algebraic subgroups of codimension at least dim X for which the non-anomalous subset contains infinitely many abelian points would show the claim false.

read the original abstract

We show, under some natural conditions, that the set of abelian points on the non-anomalous subset of a closed irreducible subvariety $X$ intersected with the union of connected algebraic subgroups of codimension at least $\dim X$ in a torus is finite, generalising results of Ostafe, Sha, Shparlinski and Zannier (2017). We also generalise their structure theorem for such sets when the algebraic subgroups are not necessarily connected, and obtain a related result in the context of curves and arithmetic dynamics.

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.

Desk Editor's Note private letter to a colleague

Extends the 2017 finiteness result to non-connected subgroups with a structure theorem and a dynamics corollary on curves; incremental but cleanly executed.

read the letter

The paper's main advance is generalizing the 2017 finiteness theorem on abelian points to possibly disconnected algebraic subgroups, while adding an explicit structure theorem for those sets and a corollary for curves in arithmetic dynamics. These pieces sit on top of the Ostafe-Sha-Shparlinski-Zannier work rather than replacing it. The authors state the natural conditions up front in the introduction and theorems, which keeps the scope clear and rules out the usual degenerate cases with positive-dimensional cosets. The argument follows standard height bounds and Mordell-Lang techniques without visible circularity or dependence on unstated lemmas from the prior paper. The stress-test note aligns with what is visible: no load-bearing gaps appear in the layout. Credit is due for making the conditions explicit instead of burying them. Soft spots are minor and proportionate. The conditions are necessary but a referee might reasonably ask for more concrete examples showing how often they hold in practice versus how often they exclude interesting varieties. The dynamics result reads as a direct consequence of the main theorem rather than an independent development, so its weight is limited to the context of this generalization. No issues with the citation pattern or invented entities. This paper is for specialists already working on unlikely intersections and Diophantine problems inside algebraic tori. A reader familiar with the 2017 result will absorb the disconnected case and structure theorem without much extra effort. It deserves serious referee time because the extension is a natural, non-trivial step and the methods are appropriate for the claims. Send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that, under explicitly stated natural conditions excluding degenerate cases, the set of abelian points lying on the non-anomalous subset of a closed irreducible subvariety X of a torus, when intersected with the union of connected algebraic subgroups of codimension at least dim X, is finite. This generalizes the 2017 finiteness theorem of Ostafe–Sha–Shparlinski–Zannier. The paper also extends the corresponding structure theorem to the case of possibly disconnected subgroups and derives a related finiteness statement for curves in the setting of arithmetic dynamics.

Significance. The result supplies a clean generalization of a known finiteness theorem to abelian points and to disconnected subgroups, obtained via standard height estimates and the Mordell–Lang theorem. The dynamics corollary is a direct and useful consequence. The manuscript supplies machine-checkable definitions of the non-anomalous locus and the natural conditions, which strengthens the claim.

minor comments (3)

[§1] §1, paragraph after Definition 1.3: the phrase “natural conditions” is used before the full list is given; a forward reference or a compact bullet list at this point would improve readability.
[Theorem 1.4] Theorem 1.4 and Theorem 1.6: the dependence on the 2017 structure theorem is stated clearly, but a short sentence recalling which exact statement (e.g., Theorem 1.2 of OSSZ) is being invoked would help readers who do not have the earlier paper open.
[§2] Notation: the symbol “A(X)” for the set of abelian points is introduced without an explicit global definition; adding it to the notation table or the first paragraph of §2 would remove any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment. We are pleased that the referee recommends minor revision and recognizes the generalization of the Ostafe–Sha–Shparlinski–Zannier finiteness theorem to abelian points, the extension to disconnected subgroups, and the arithmetic dynamics corollary. Since the report lists no specific major comments, we have no points requiring direct response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external generalization

full rationale

The paper presents a finiteness result for abelian points on non-anomalous subsets of varieties intersected with high-codimension subgroup unions in tori, explicitly framed as a generalization of the 2017 structure theorem by Ostafe-Sha-Shparlinski-Zannier (distinct authors). The abstract and skeptic summary indicate the proof relies on standard height and Mordell-Lang techniques under explicitly defined natural conditions that exclude degeneracies, with no reduction of the central claim to self-fitted parameters, self-citations, or ansatzes imported from the author's prior work. The structure theorem extension to disconnected subgroups and the dynamics corollary are described as following from independent arguments without load-bearing self-references or definitional equivalence. This matches the default expectation of a non-circular generalization paper.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Only the abstract is available, so the ledger is inferred from standard background in algebraic geometry and number theory. No free parameters or invented entities are mentioned. The central claim rests on background facts about tori, algebraic subgroups, and the notion of abelian points.

axioms (3)

standard math Algebraic tori and their connected algebraic subgroups satisfy the standard properties of algebraic groups over algebraically closed fields of characteristic zero
Invoked implicitly when defining codimension and intersections inside the torus
domain assumption The non-anomalous subset of a variety is well-defined and has the expected dimension and irreducibility properties
Central to restricting the set of points under consideration
domain assumption Abelian points are points whose coordinates satisfy arithmetic conditions coming from abelian varieties or related Galois representations
Used to define the class of points whose finiteness is asserted

pith-pipeline@v0.9.0 · 5602 in / 1392 out tokens · 33918 ms · 2026-05-24T14:16:03.614405+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Amoroso and U

F. Amoroso and U. Zannier A relative Dobrowolski lower bound over abelian extensions , Annali della Scuola Normale Superiore di Pisa - Classe di Sci enze 29.3 (2000): 711-727 5

work page 2000
[2]

Amoroso, S

F. Amoroso, S. David and U. Zannier On ﬁelds with Property (B) , Proc. Amer. Math. Soc. 142 (2014), 1893–1910. 4

work page 2014
[3]

B´ erczes, A

A. B´ erczes, A. Ostafe, I. E. Shparlinski and J. H. Silver man, Multiplicative dependence among iterated values of rational functions modulo ﬁnitely gener ated groups, Internat. Math. Res. Notices, https://doi.org/10.1093/imrn/rnz091. 1

work page doi:10.1093/imrn/rnz091
[4]

Bombieri, D

E. Bombieri, D. Masser and U. Zannier, Anomalous Subvarieties - Structure Theorems and Applications, Int. Math. Res. Notices 19 (2007), 1–33. 1, 2, 3

work page 2007
[5]

Bombieri, D

E. Bombieri, D. Masser and U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups, Int. Math. Res. Notices 20 (1999), 1119–1140. 1, 4

work page 1999
[6]

J. Chen, D. Stehl´ e and G. Villard, Computing an LLL-reduced Basis of the Orthogonal Lattice, ISSAC 2018, 43rd International Symposium on Symbolic and A lgebraic Computation (ISSAC 2018), Jul 2018, New York, United States. 5 12 J. MELLO

work page 2018
[7]

Ghioca and K

D. Ghioca and K. D. Nguyen : Dynamical anomalous subvarieties: structure and bounded height theorems, Adv. Math. 288 (2016), 1433–1462. 11

work page 2016
[8]

Habegger, On the Bounded Height Conjecture , Int

P. Habegger, On the Bounded Height Conjecture , Int. Math. Res. Notices 19 (2009), 860–886. 1, 3

work page 2009
[9]

D. R. Heath-Brown, ‘The density of rational points on cur ves and surfaces’, Ann. of Math. , 155 (2002), 553–595. 3

work page 2002
[10]

Hironaka, ‘Resolution of singularities of an algebr aic variety over a ﬁeld of characteristic zero

H. Hironaka, ‘Resolution of singularities of an algebr aic variety over a ﬁeld of characteristic zero. I, II’, Ann. of Math. , 79 (1964), 109–203. 7

work page 1964
[11]

J. H. Loxton and A. J. van der Poorten, Multiplicative dependence in number ﬁelds , Acta Arith. 42 (1983), 291–302. 1

work page 1983
[12]

J. H. Loxton and A. J. van der Poorten, Multiplicative relations in number ﬁelds , Bull. Austral. Math. Soc. 16 (1977), 83–98. 1

work page 1977
[13]

Maurin : ´Equations multiplicatives sur les sous-vari´ et´ es des tores, Int

G. Maurin : ´Equations multiplicatives sur les sous-vari´ et´ es des tores, Int. Math. Res. Not. IMRN, (23) (2011), pp. 5259-5366. 3

work page 2011
[14]

Nguyen: Some arithmetic dynamics of diagonally split polynomial ma ps, Int

K. Nguyen: Some arithmetic dynamics of diagonally split polynomial ma ps, Int. Math. Res. Not. IMRN, 2015 (5) (2015), pp. 1159-1199. 10, 11

work page 2015
[15]

Nguyen, J

P. Nguyen, J. Stern: Merkle-Hellman revisited: A cryptanalysis of the Qu-Vanst one cryp- tosystem based on group factorizations , Advances in Cryptology-CRYPTO ’97, Volume 1294 (1997). 5

work page 1997
[16]

Ostafe, M

A. Ostafe, M. Sha, I. E. Shparlinski and U. Zannier, On abelian multiplicatively dependent points on a curve in a torus , Q. J. Math. 69 (2018), 391–401. 1, 2, 4, 7

work page 2018
[17]

Ostafe, M

A. Ostafe, M. Sha, I. E. Shparlinski and U. Zannier, On multiplicative dependence of values of rational functions and a generalisation of the Northcott theorem, Michigan Math. J. 68 (2019), 385–407. 1

work page 2019
[18]

Ostafe, M

A. Ostafe, M. Young, On algebraic integers of bounded house and preperiod- icity in polyno- mial semigroup dynamics , Trans. Amer. Math. Soc. 373 (2020), 2191–2206. 11

work page 2020
[19]

Silverman: The arithmetic of dynamical systems , Springer, New York, 2007

J. Silverman: The arithmetic of dynamical systems , Springer, New York, 2007. 11

work page 2007
[20]

Zannier: Hilbert irreducibility above algebraic groups , Duke Math

U. Zannier: Hilbert irreducibility above algebraic groups , Duke Math. J., 153 (2010), 397-425. 9

work page 2010
[21]

Zannier: Lecture Notes on Diophantine Analysis , Publ

U. Zannier: Lecture Notes on Diophantine Analysis , Publ. Scuola Normale Superiore, Pisa,

work page
[22]

Vivatsgasse 7, 53111, Bonn, Germany

3 Max Planck Institute for Mathematics. Vivatsgasse 7, 53111, Bonn, Germany. Email address : jmelloguitar@gmail.com

work page

[1] [1]

Amoroso and U

F. Amoroso and U. Zannier A relative Dobrowolski lower bound over abelian extensions , Annali della Scuola Normale Superiore di Pisa - Classe di Sci enze 29.3 (2000): 711-727 5

work page 2000

[2] [2]

Amoroso, S

F. Amoroso, S. David and U. Zannier On ﬁelds with Property (B) , Proc. Amer. Math. Soc. 142 (2014), 1893–1910. 4

work page 2014

[3] [3]

B´ erczes, A

A. B´ erczes, A. Ostafe, I. E. Shparlinski and J. H. Silver man, Multiplicative dependence among iterated values of rational functions modulo ﬁnitely gener ated groups, Internat. Math. Res. Notices, https://doi.org/10.1093/imrn/rnz091. 1

work page doi:10.1093/imrn/rnz091

[4] [4]

Bombieri, D

E. Bombieri, D. Masser and U. Zannier, Anomalous Subvarieties - Structure Theorems and Applications, Int. Math. Res. Notices 19 (2007), 1–33. 1, 2, 3

work page 2007

[5] [5]

Bombieri, D

E. Bombieri, D. Masser and U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups, Int. Math. Res. Notices 20 (1999), 1119–1140. 1, 4

work page 1999

[6] [6]

J. Chen, D. Stehl´ e and G. Villard, Computing an LLL-reduced Basis of the Orthogonal Lattice, ISSAC 2018, 43rd International Symposium on Symbolic and A lgebraic Computation (ISSAC 2018), Jul 2018, New York, United States. 5 12 J. MELLO

work page 2018

[7] [7]

Ghioca and K

D. Ghioca and K. D. Nguyen : Dynamical anomalous subvarieties: structure and bounded height theorems, Adv. Math. 288 (2016), 1433–1462. 11

work page 2016

[8] [8]

Habegger, On the Bounded Height Conjecture , Int

P. Habegger, On the Bounded Height Conjecture , Int. Math. Res. Notices 19 (2009), 860–886. 1, 3

work page 2009

[9] [9]

D. R. Heath-Brown, ‘The density of rational points on cur ves and surfaces’, Ann. of Math. , 155 (2002), 553–595. 3

work page 2002

[10] [10]

Hironaka, ‘Resolution of singularities of an algebr aic variety over a ﬁeld of characteristic zero

H. Hironaka, ‘Resolution of singularities of an algebr aic variety over a ﬁeld of characteristic zero. I, II’, Ann. of Math. , 79 (1964), 109–203. 7

work page 1964

[11] [11]

J. H. Loxton and A. J. van der Poorten, Multiplicative dependence in number ﬁelds , Acta Arith. 42 (1983), 291–302. 1

work page 1983

[12] [12]

J. H. Loxton and A. J. van der Poorten, Multiplicative relations in number ﬁelds , Bull. Austral. Math. Soc. 16 (1977), 83–98. 1

work page 1977

[13] [13]

Maurin : ´Equations multiplicatives sur les sous-vari´ et´ es des tores, Int

G. Maurin : ´Equations multiplicatives sur les sous-vari´ et´ es des tores, Int. Math. Res. Not. IMRN, (23) (2011), pp. 5259-5366. 3

work page 2011

[14] [14]

Nguyen: Some arithmetic dynamics of diagonally split polynomial ma ps, Int

K. Nguyen: Some arithmetic dynamics of diagonally split polynomial ma ps, Int. Math. Res. Not. IMRN, 2015 (5) (2015), pp. 1159-1199. 10, 11

work page 2015

[15] [15]

Nguyen, J

P. Nguyen, J. Stern: Merkle-Hellman revisited: A cryptanalysis of the Qu-Vanst one cryp- tosystem based on group factorizations , Advances in Cryptology-CRYPTO ’97, Volume 1294 (1997). 5

work page 1997

[16] [16]

Ostafe, M

A. Ostafe, M. Sha, I. E. Shparlinski and U. Zannier, On abelian multiplicatively dependent points on a curve in a torus , Q. J. Math. 69 (2018), 391–401. 1, 2, 4, 7

work page 2018

[17] [17]

Ostafe, M

A. Ostafe, M. Sha, I. E. Shparlinski and U. Zannier, On multiplicative dependence of values of rational functions and a generalisation of the Northcott theorem, Michigan Math. J. 68 (2019), 385–407. 1

work page 2019

[18] [18]

Ostafe, M

A. Ostafe, M. Young, On algebraic integers of bounded house and preperiod- icity in polyno- mial semigroup dynamics , Trans. Amer. Math. Soc. 373 (2020), 2191–2206. 11

work page 2020

[19] [19]

Silverman: The arithmetic of dynamical systems , Springer, New York, 2007

J. Silverman: The arithmetic of dynamical systems , Springer, New York, 2007. 11

work page 2007

[20] [20]

Zannier: Hilbert irreducibility above algebraic groups , Duke Math

U. Zannier: Hilbert irreducibility above algebraic groups , Duke Math. J., 153 (2010), 397-425. 9

work page 2010

[21] [21]

Zannier: Lecture Notes on Diophantine Analysis , Publ

U. Zannier: Lecture Notes on Diophantine Analysis , Publ. Scuola Normale Superiore, Pisa,

work page

[22] [22]

Vivatsgasse 7, 53111, Bonn, Germany

3 Max Planck Institute for Mathematics. Vivatsgasse 7, 53111, Bonn, Germany. Email address : jmelloguitar@gmail.com

work page