Equidistribution for semiabelian varieties over number fields

Wei Xue

REVIEW 1 major objections 4 minor 36 references

Equidistribution for semiabelian varieties over number fields

T0 review · 1 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read The paper proves that generic small sequences on a compactified semiabelian variety equidistribute at every place when the toric metric is big, T-effective, monocritical, and arithmetically T-effective, and that the…

desk verdict A careful, honest extension of Kühne's semiabelian equidistribution to a controlled non-canonical toric metric class, but the load-bearing error-scaling Lemma 35 is asserted rather than proved, and the stress-test computation suggests it may be wrong. read the letter →

arxiv 2608.08262 v1 pith:K3SOBM72 submitted 2026-08-08 math.NT math.AG

Wei Xue This is my paper · ORCID

classification math.NTmath.AG MSC 11G5014G4014M2514K15

keywords equidistributionsemiabelianvarietiestoricmetrizeddivisorsmonocriticalmetricsarithmeticT-effectivityBogomolovtheoremadeliclinebundlesheights

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 aims to prove that small algebraic points on a semiabelian variety with a toric compactification spread out uniformly at every place, even when the metric on the toric part is not the canonical one. The main theorem says this holds whenever the toric metrized divisor is big, T-effective, monocritical, and arithmetically T-effective; in the quasi-canonical case the limit measure is the explicitly normalized mixed curvature measure of the toric bundle and the pulled-back abelian line bundle. The paper also proves the corresponding Bogomolov statement: a subvariety whose essential minimum equals that of the whole variety must be special, meaning a translate of a connected algebraic subgroup by a torsion point, with the converse holding under a special-point hypothesis. The payoff is that earlier canonical semiabelian equidistribution is extended to a controlled class of non-canonical toric metrics, and the proof works by compressing the metric toward a quasi-canonical model along explicit paths.

What carries the argument

The load-bearing mechanism is the explicit compression path $\overline{D}_m = m^{-1}[m]^*\overline{D}$ for a quasi-canonical metric, whose local support functions take the normal form $\Psi_{\overline{D}_m,v}(u) = \Psi_{\overline{D}}(u - u_v/m) - \gamma_v/m$, together with the auxiliary $n$-division tower $G_n$ on which height, intersection, and measure estimates are made. The crucial scaling lemma (Lemma 35) asserts that after normalization by $L_n^d$, the quadratic core of the error terms retains at most one explicit factor $n$, and after horizontal compression by $[m]$ at most one additional factor $m$, so the compressed quadratic errors are $O(n)$ and $O(nm)$. This dimension-count bookkeeping feeds every later compressed estimate. The monocritical case is then reduced to the quasi-canonical case by replacing the metric with $\Psi_{\overline{D}',v}(z) = \Psi_{\overline{D}}(z - u_v)$, where $(u_v)_v$ is the critical tropical vector; this replacement preserves smallness of sequences and forces the limiting valuation measure to be the Dirac mass at the critical vector.

What would settle it

Write out the first-variation error expansion explicitly for the split example $G = \mathbb{G}_m^2 \times E$ with $\mathbb{P}^1 \times \mathbb{P}^1$ compactification and a non-canonical monocritical metric, and check whether the normalized quadratic error term is bounded by $O(nm)$ as Lemma 57 claims; an explicit computation showing $O(n^2m)$ or $O(nm^2)$ would break the equidistribution proof.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.3: for a semiabelian variety $G$ over a number field $K$ with split toric part of dimension $t$ and abelian quotient $A$ of dimension $g$, let $\mathbb{G}$ be the compactification attached to a proper toric variety $X_\Sigma$, and let $\overline{D}$ be a toric metrized $\mathbb{R}$-divisor on $X_\Sigma$. If $\overline{D}$ is big, $T$-effective, monocritical, and arithmetically $T$-effective, then the adelic line bundle $\overline{L} = \mathcal{G}(\overline{D}) \otimes \pi^*\overline{N}$ has the $v$-adic equidistribution property at every place $v$: every generic $\overline{L}$-small sequence of $K$-points has $v$-adic Galois orbit measures converging weakly, with limit equal, in the quasi-canonical case, to $\hat{c}_1(\mathcal{G}(\overline{D})_v)^t \wedge \hat{c}_1(\pi^*\overline{N}_v)^g$ divided by $\mathcal{G}(\overline{D})^t(\pi^*N)^g$. Theorem 1.4 draws the Bogomolov consequence: under the same hypotheses, an irreducible subvariety $X$ with $\mu^{\mathrm{ess}}_{\overline{L}}(X) = \mu^{\mathrm{ess}}_{\overline{L}}(\mathbb{G})$ is special, and the converse holds when $\mathbb{G}(K)$ has $\overline{L}$-special points. The proof isolates the asymptotic estimates used in the canonical semiabelian case and reinterprets them as estimates along explicit compression paths, giving a comparison mechanism between canonical, quasi-canonical, and more general toric metrics.

Load-bearing premise

The proof depends on a dimension-count estimate saying that the quadratic error terms, after rescaling by the top self-intersection, grow at most linearly in the division parameter $n$ and at most linearly in the compression parameter $m$; if the true growth were faster, the errors would not vanish.

Editorial extensions

If this is right

Every generic $\overline{L}$-small sequence in $\mathbb{G}(\overline{K})$ has $v$-adic Galois orbit measures converging weakly at every place $v$ under the stated conditions on $\overline{D}$.
When the toric metric is additionally quasi-canonical, the limiting measure is the normalized mixed Chern measure $\hat{c}_1(\mathcal{G}(\overline{D})_v)^t \wedge \hat{c}_1(\pi^*\overline{N}_v)^g / (\mathcal{G}(\overline{D})^t(\pi^*N)^g)$.
Equality of essential minima, $\mu^{\mathrm{ess}}_{\overline{L}}(X) = \mu^{\mathrm{ess}}_{\overline{L}}(\mathbb{G})$, forces $X$ to be special, and the converse holds when $\mathbb{G}(K)$ has $\overline{L}$-special points.
Strict quasi-canonical equidistribution holds on the minimal translates that arise in the Bogomolov restriction argument, supplying the non-circular upgrade used in the proof.

Reading between the lines

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

The compression method suggests a general recipe for other arithmetic equidistribution problems: if a reference metric is approached by a one-parameter family of metrics with controlled error growth after normalization, equidistribution transfers; testing this on semipositive toric metrics that are not monocritical would show whether the unique-critical-vector condition can be relaxed to a finite
The explicit parameter choice $m(n) = \lfloor n^a\rfloor$, $\lambda = n^{-(3+a)/2}$, $0 < a < 1$, predicts convergence rates $O(n^{-(1-a)/2})$; a numerical check on a split example such as $\mathbb{G}_m^2 \times E$ could test whether this rate is sharp or merely an artifact of the proof.
The paper deliberately leaves arbitrary semipositive toric metrics untouched; if the monocritical condition fails, the limiting measure might be a mixture of Dirac masses at several critical vectors, and the Bogomolov implication would need a separate argument.

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

Two things to know. First, this paper is not padding: the compression-path mechanism and the monocritical/arithmetically T-effective metric class are genuinely new relative to Kühne, BGPS, and Yuan–Zhang, and the paper is unusually careful about what it does and does not prove. Second, the main theorem rests on Lemma 35, and the proof of that lemma is not really a proof. It is a dimension count that bounds the number of toric divisor classes, not the metric slopes that carry the n-dependence. The stress-test model on (P^1)^2 is concrete and worrying: f_n(u,v)=f(nu,nv) multiplies wall weights by n in each coordinate, so the quadratic core F_n^2 · L^{d-1}/L^d should grow like n^2, not n. If that computation is right, Lemmas 36, 57–59 and the §6.4 parameter choice collapse, and Theorem 1.3 is not established.

What the paper does well deserves credit. The source ledgers are honest and specific, and Remark 61 explicitly flags the generic/strict distinction and refuses to import strict equidistribution without circularity. The paper also states its scope clearly: it is not a new proof of the canonical theorem and not a theorem for all semipositive metrics. Proposition 4.27's valuation/height package and the arithmetic T-effectivity definition look coherent, and the reduction from monocritical to quasi-canonical in Section 8 is a sensible use of the BGPS KR machinery.

The other soft spot is Proposition 96, which asserts that Kühne's local-trivialization argument extends verbatim to arbitrary toric compactifications. Kühne worked with (P^1)^t; the extension to a general fan requires the corner-current package of §9.4–9.5. There are several "this is exactly Kühne's computation" transitions without a written check, so even granting Lemma 35, I would want the verbatim claim verified line by line. The final restriction package is truncated in the input, so I could not assess the last step of the Bogomolov argument.

Who is this for? Arithmetic geometers working on equidistribution, heights, and Bogomolov. A serious referee should definitely see it, because the overall architecture is important and the gap is a specific, checkable estimate. As it stands, I would not cite Theorem 1.3 in my own work, and the paper needs heavy revision before I would trust it. But this is a conditional reject, not a dismiss: the framework is sound enough that fixing or replacing Lemma 35 could make the main theorems true.

Recommendation: send to peer review, with a referee who is explicitly asked to verify Lemma 35 and Proposition 96.

Referee Report

1 major / 4 minor

Summary. The paper proves generic equidistribution for adelic line bundles of the form L = G(D) ⊗ π*N on toric compactifications of semiabelian varieties over number fields, where D is a big T-effective toric divisor equipped with a semipositive, monocritical, arithmetically T-effective toric metric. It further derives a Bogomolov rigidity theorem for the same metric class. The proof splits into canonical, quasi-canonical (via explicit compression paths), and monocritical stages, then transports Kühne's local-trivialization and difference-morphism argument to general toric compactifications for the Bogomolov application.

Significance. If correct, this extends Kühne's canonical semiabelian equidistribution theorem to a class of non-canonical toric metrics, and gives the corresponding Bogomolov rigidity. The manuscript is unusually careful with source attribution, explicitly flags the generic/strict distinction, and provides a detailed local-trivialization construction at all places. The main theorems are crisp and falsifiable. The paper also ships, at the level of a written proof, a systematic source ledger for the external inputs it uses.

major comments (1)

[§5.3, Lemma 35 (used in Lemmas 36, 57–59, §6.4)] The proof of Lemma 35 asserts that after normalization by L_n^d the quadratic core of the error terms retains at most one explicit factor n (and at most one further factor m after compression), but this is justified by a dimension count rather than a written derivation. The dimension count bounds the number of algebraic divisor classes in a top intersection; it does not control the size of the metric slopes of the pulled-back test metric F_n = φ_n^*O(f), which enter through the local potentials of f. In the model case G = G_m^2, X = (P^1)^2, D = O(1,1), a toric test metric f with walls along the two coordinate directions gives f_n(u,v) = f(nu,nv), so the r = 2 term F_n^2·L_n^{d-1}/L_n^d is of order n^2 times the mixed volume of the dual cells, not O(n). If this model computation is representative, then Lemmas 36, 57, 58, and 59 inherit the wrong scaling, and the parameter choice m(n) = n^{1/2}, λ = n^{-7/4} in §6.4 no longer makes the Minkowski error vanish. Since these estimates are the only control on the quadratic error in the variation argument, Theorems 1.3 and 1.4 are not established unless Lemma 35 is proved or replaced by a correct scaling estimate.

minor comments (4)

[Title page] There are several spacing typos in the title and headers, e.g. 'V arieties' and occasional 'Kuhne' instead of 'Kühne'.
[Throughout] Auxiliary statements are numbered inconsistently: some are 'Proposition 12', 'Lemma 13', while others are 'Theorem 4.1', 'Proposition 4.17'. This makes cross-referencing needlessly difficult.
[§5.3, Lemma 35] The term 'quadratic core' is never defined precisely. Since the lemma is load-bearing, the statement should formalize the exact intersection products and normalizations involved.
[§9.5, Proposition 96] The phrase 'extends verbatim' is too strong given that the proof supplies nontrivial replacements for Kühne's Lemmas 18–20; suggesting a black-box transfer obscures the actual verification performed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are derived from external benchmarks (Kühne, Yuan–Zhang, BGPS, Yuan/Ikoma bigness) and internal compression/replacement reductions that do not assume the target results.

full rationale

The paper's central claim, generic v-adic equidistribution for semipositive, monocritical, arithmetically T-effective toric metrics on semiabelian compactifications, is not equivalent by construction to any fitted input or to a self-citation chain. The proof proceeds by reducing to the canonical metric case via Kühne's external estimates, then to quasi-canonical metrics by explicit compression, and finally to monocritical metrics by the quasi-canonical replacement of Proposition 71 together with BGPS toric measure theory. Each of these is an external source theorem or an internal reduction whose hypotheses do not include the target equidistribution. The only place where circularity is explicitly risked is the generic/strict upgrade, and the paper addresses this directly in Remark 61, stating that strict quasi-canonical equidistribution cannot be imported from the paper's own Bogomolov theorem without circularity; it instead obtains the strict statement from an independent quasi-canonical Bogomolov theorem, Theorem 76, through Proposition 101. Lemma 35's asserted O(n) error scaling is a potential correctness gap, not a circular step: the estimate is not used to define the quantities it bounds, and a wrong bound would invalidate the proof rather than making the conclusion tautological. Self-citations to Kühne, BGPS, Yuan–Zhang, and Yuan/Ikoma are load-bearing but are citations to independent, externally established results; Remark 18 and Remark 34 explicitly record which parts are imported and which are new. No equation in the paper reduces to a fitted parameter renamed as a prediction, nor does any uniqueness theorem from the authors' prior work force the main choice. Therefore the derivation is self-contained relative to its cited external benchmarks, and no circularity is present.

Assumptions & free parameters 4 free parameters · 8 assumptions · 0 invented entities

The central claims rest on a large body of established results; the only new restriction is the arithmetically T-effective domain condition. No new physical or geometric entities are introduced.

free parameters (4)

Compression path exponent m(n)=floor(n^a), 0<a<1 = n^{1/2} in the explicit choice a=1/2
Auxiliary error-balancing parameter chosen in Section 6.4; the theorem needs existence of some a, no particular value.
Minkowski perturbation lambda_n = n^{-7/4} for a=1/2
Chosen so that |lambda| <= (n m(n))^{-1} and all normalized error terms vanish as n grows.
Approximation parameter epsilon_n = n^{-1/2} for a=1/2
Auxiliary parameter in the small-sequence argument, set to |lambda| n^{-(1-a)/2}.
Constant vertical correction kappa_n = O(n^{-2})
Restores horizontal semipositivity in Propositions 32 and 50; existence is proved, not fitted to data.

assumptions (8)

standard math Kühne's canonical semiabelian equidistribution and local trivialization package
Used as the baseline for the canonical case and for the transport package in Proposition 96.
standard math Yuan-Zhang adelic line bundle and equidistribution framework
Provides definitions of heights, essential minima, and the quasi-canonical theorem used as benchmark.
standard math BGPS toric measure rigidity and monocritical classification
Supplies the Kantorovich-Rubinstein package, critical vector, and atomic tropical pushforwards used in Sections 8 and 9.
standard math Yuan/Ikoma arithmetic bigness theorem in the form of Kühne's Lemma 7 and Lemma 17
Underpins the quadratic arithmetic volume comparison in Proposition 37, Lemma 43, and Lemma 59.
standard math Raynaud-Bosch-Lütkebohmert uniformization, p-adic theta functions and Berkovich skeleton
Used in Theorem 4.1 to extend local trivializations to non-archimedean places.
standard math Toric successive minima formula
Used in Lemma 27 and Proposition 4.28 for the equality of essential and absolute minima on the torus.
standard math Zhang's Bogomolov theorem for abelian varieties
Used in the base reduction step of Proposition 100.
domain assumption The metric D is arithmetically T-effective in the sense of Definition 4.20
This is a hypothesis of the main theorems, not proven automatic for all monocritical metrics; it is needed for the height lower bound and the minima formula.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Equidistribution for semiabelian varieties over number fields." pith.science (2026). https://pith.science/paper/K3SOBM72

@misc{pith2026260808262,
  author       = {Pith},
  title        = {Pith review of: Equidistribution for semiabelian varieties over number fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K3SOBM72}},
  note         = {Machine review of arXiv:2608.08262}
}

read the original abstract

K{\"u}hne established equidistribution for canonical adelic line bundles on semiabelian varieties, a setting which need not lie in the quasi-canonical range of Yuan-Zhang's theorem. We study adelic line bundles on semiabelian compactifications whose toric part is governed by a toric metrized divisor. The main unconditional result is generic equidistribution when the toric metric is monocritical and arithmetically T-effective. The proof isolates the asymptotic estimates in K{\"u}hne's argument and reinterprets them as estimates along explicit compression paths. This gives a comparison mechanism between canonical, quasi-canonical, and more general toric metrics. For the Bogomolov application, the quasi-canonical case is handled by a K{\"u}hne local-trivialization transport package. After fixing a single theta-factor convention for the local trivializations, the proof checks the Picard-zero theta factors under K{\"u}hne's operations and obtains the Bogomolov theorem for the metric class treated in this paper, namely the monocritical and arithmetically T-effective toric metrics, through the quasi-canonical replacement argument.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 33 canonical work pages

[1]

Points of small height on semiabelian varieties.J

Lars K¨ uhne. Points of small height on semiabelian varieties.J. Eur. Math. Soc., 24(6):2077–2131, 2022

work page 2022
[2]

Equidistribution in families of abelian varieties and uniformity, 2021

Lars K¨ uhne. Equidistribution in families of abelian varieties and uniformity, 2021. arXiv:2101.10272

arXiv 2021
[3]

Boundedness of the successive minima on arithmetic varieties.J

Hideaki Ikoma. Boundedness of the successive minima on arithmetic varieties.J. Algebraic Geom., 22(2):249–302, 2013

work page 2013
[4]

G´ eom´ etrie d’Arakelov et hauteurs canoniques sur des vari´ et´ es semi-ab´ eliennes

Antoine Chambert-Loir. G´ eom´ etrie d’Arakelov et hauteurs canoniques sur des vari´ et´ es semi-ab´ eliennes. Math. Ann., 314:381–401, 1999

work page 1999
[5]

Springer-Verlag, 1982

Serge Lang.Introduction to Algebraic and Abelian Functions. Springer-Verlag, 1982

work page 1982
[6]

Small points and adelic metrics.J

Shou-Wu Zhang. Small points and adelic metrics.J. Algebraic Geom., 4:281–300, 1995

work page 1995
[9]

The bounded height conjecture for semiabelian varieties.Compos

Lars K¨ uhne. The bounded height conjecture for semiabelian varieties.Compos. Math., 156(7):1405–1456, 2020

work page 2020
[10]

Successive minima and asymptotic slopes in Arakelov geometry.Compos

Fran¸ cois Balla¨ y. Successive minima and asymptotic slopes in Arakelov geometry.Compos. Math., 157(8):1302–1339, 2020

work page 2020

Show all 36 references

[11]

Local and canonical heights of subvarieties.Ann

Walter Gubler. Local and canonical heights of subvarieties.Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2(4):711–760, 2003

2003
[12]

Metrics, measures and heights

Jos´ e Ignacio Burgos Gil, Patrice Philippon, and Mart ´ ın Sombra.Arithmetic geometry of toric varieties. Metrics, measures and heights. Ast´ erisque 360, Soci´ et´ e Math´ ematique de France, 2014

2014
[13]

Berkovich.Spectral Theory and Analytic Geometry over Non-Archimedean Fields

Vladimir G. Berkovich.Spectral Theory and Analytic Geometry over Non-Archimedean Fields. Math. Surveys Monogr. 33, Amer. Math. Soc., 2012

2012
[14]

Non-Archimedean and tropical theta functions.Math

Tyler Foster, Joseph Rabinoff, Farbod Shokrieh, and Alejandro Soto. Non-Archimedean and tropical theta functions.Math. Ann., 372(3–4):891–914, 2017

2017
[15]

The distribution of Galois orbits of points of small height in toric varieties.Amer

Jos´ e Ignacio Burgos Gil, Patrice Philippon, Juan Rivera-Letelier, and Mart ´ ın Sombra. The distribution of Galois orbits of points of small height in toric varieties.Amer. J. Math., 141(2):309–381, 2019

2019
[16]

Arithmetic positivity on toric varieties.J

Jos´ e Ignacio Burgos Gil, Atsushi Moriwaki, Patrice Philippon, and Mart ´ ın Sombra. Arithmetic positivity on toric varieties.J. Algebraic Geom., 25(4):201–272, 2016

2016
[17]

Successive minima of toric height func- tions.Ann

Jos´ e Ignacio Burgos Gil, Patrice Philippon, and Mart ´ ın Sombra. Successive minima of toric height func- tions.Ann. Inst. Fourier (Grenoble), 65(5):2145–2197, 2015

2015
[18]

Adelic line bundles on quasi-projective varieties, 2023

Xinyi Yuan and Shou-Wu Zhang. Adelic line bundles on quasi-projective varieties, 2023. arXiv:2105.13587

2023 arXiv
[19]

William Fulton.Introduction to Toric Varieties. Ann. of Math. Stud. 131, Princeton Univ. Press, 1993. 82 WEI XUE

1993
[20]

Springer, 2013

Serge Lang.Fundamentals of Diophantine Geometry. Springer, 2013

2013
[21]

Serge Lang.Introduction to Algebraic and Abelian Functions. Grad. Texts in Math. 89, Springer, 1982

1982
[22]

Points de petite hauteur sur les vari´ et´ es semi-ab´ eliennes.Ann

Antoine Chambert-Loir. Points de petite hauteur sur les vari´ et´ es semi-ab´ eliennes.Ann. Sci.´Ecole Norm. Sup´ er. (4), 33(6):789–821, 1999

1999
[23]

Mesures et ´ equidistribution sur les espaces de Berkovich.J

Antoine Chambert-Loir. Mesures et ´ equidistribution sur les espaces de Berkovich.J. Reine Angew. Math., 595:215–235, 2006

2006
[24]

Arithmetic on algebraic varieties.Ann

Andr´ e Weil. Arithmetic on algebraic varieties.Ann. of Math. (2), 53(3):412–444, 1951

1951
[25]

Quasi-fonctions et hauteurs sur les vari´ et´ es ab´ eliennes.Ann

Andr´ e N´ eron. Quasi-fonctions et hauteurs sur les vari´ et´ es ab´ eliennes.Ann. of Math. (2), 82(2):249–331, 1965

1965
[26]

Endlichkeitss¨ atze f¨ ur abelsche Variet¨ aten ¨ uber Zahlk¨ orpern.Invent

Gerd Faltings. Endlichkeitss¨ atze f¨ ur abelsche Variet¨ aten ¨ uber Zahlk¨ orpern.Invent. Math., 73(3):349–366, 1983

1983
[27]

Sous-vari´ et´ es d’une vari´ et´ e ab´ elienne et points de torsion

Michel Raynaud. Sous-vari´ et´ es d’une vari´ et´ e ab´ elienne et points de torsion. InArithmetic and Geometry: Papers Dedicated to I.R. Shafarevich Volume I: Arithmetic, pages 327–352. Birkh¨ auser, 1983

1983
[28]

Positivit´ e et discr´ etion des points alg´ ebriques des courbes.Ann

Emmanuel Ullmo. Positivit´ e et discr´ etion des points alg´ ebriques des courbes.Ann. of Math. (2), 147(1):167–179, 1996

1996
[29]

Equidistribution of small points on abelian varieties.Ann

Shou-Wu Zhang. Equidistribution of small points on abelian varieties.Ann. of Math. (2), 147(1):159–165, 1998

1998
[30]

´Equir´ epartition des petits points.Invent

Lucien Szpiro, Emmanuel Ullmo, and Shou-Wu Zhang. ´Equir´ epartition des petits points.Invent. Math., 127(2):337–347, 1997

1997
[31]

Big line bundles over arithmetic varieties.Invent

Xinyi Yuan. Big line bundles over arithmetic varieties.Invent. Math., 173(3):603–649, 2006

2006
[32]

Uniformity in Mordell–Lang for curves.Ann

Vesselin Dimitrov, Ziyang Gao, and Philipp Habegger. Uniformity in Mordell–Lang for curves.Ann. of Math. (2), 194(1):237–298, 2021

2021
[33]

A consequence of the relative Bogomolov conjecture,

Vesselin Dimitrov, Ziyang Gao, and Philipp Habegger. A consequence of the relative Bogomolov conjecture,
[34]

The relative Manin–Mumford conjecture, 2023

Ziyang Gao and Philipp Habegger. The relative Manin–Mumford conjecture, 2023. arXiv:2303.05045

2023 arXiv
[35]

Degenerating abelian varieties.Topology, 30(4):653–698, 1991

Siegfried Bosch and Werner L¨ utkebohmert. Degenerating abelian varieties.Topology, 30(4):653–698, 1991

1991
[36]

Approximation of adelic divisors and equidistribution of small points,

Fran¸ cois Balla¨ y and Mart ´ ın Sombra. Approximation of adelic divisors and equidistribution of small points,
[37]

Arakelov geometry of toric bundles: Okounkov bodies and BKK, 2024

Nuno Hultberg. Arakelov geometry of toric bundles: Okounkov bodies and BKK, 2024. arXiv:2412.04169

2024 arXiv
[38]

James S. Milne. Abelian Varieties (v2.00), 2008. Available at www.jmilne.org/math/

2008

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Points of small height on semiabelian varieties.J

Lars K¨ uhne. Points of small height on semiabelian varieties.J. Eur. Math. Soc., 24(6):2077–2131, 2022

work page 2022

[2] [2]

Equidistribution in families of abelian varieties and uniformity, 2021

Lars K¨ uhne. Equidistribution in families of abelian varieties and uniformity, 2021. arXiv:2101.10272

arXiv 2021

[3] [3]

Boundedness of the successive minima on arithmetic varieties.J

Hideaki Ikoma. Boundedness of the successive minima on arithmetic varieties.J. Algebraic Geom., 22(2):249–302, 2013

work page 2013

[4] [4]

G´ eom´ etrie d’Arakelov et hauteurs canoniques sur des vari´ et´ es semi-ab´ eliennes

Antoine Chambert-Loir. G´ eom´ etrie d’Arakelov et hauteurs canoniques sur des vari´ et´ es semi-ab´ eliennes. Math. Ann., 314:381–401, 1999

work page 1999

[5] [5]

Springer-Verlag, 1982

Serge Lang.Introduction to Algebraic and Abelian Functions. Springer-Verlag, 1982

work page 1982

[6] [6]

Small points and adelic metrics.J

Shou-Wu Zhang. Small points and adelic metrics.J. Algebraic Geom., 4:281–300, 1995

work page 1995

[7] [9]

The bounded height conjecture for semiabelian varieties.Compos

Lars K¨ uhne. The bounded height conjecture for semiabelian varieties.Compos. Math., 156(7):1405–1456, 2020

work page 2020

[8] [10]

Successive minima and asymptotic slopes in Arakelov geometry.Compos

Fran¸ cois Balla¨ y. Successive minima and asymptotic slopes in Arakelov geometry.Compos. Math., 157(8):1302–1339, 2020

work page 2020