Unlikely intersections in Shimura varieties and beyond: a survey

Christopher Daw

REVIEW 2 major objections 5 minor 165 references

Unlikely intersections in Shimura varieties and beyond: a survey

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read André–Oort is now fully proved, according to this survey of Shimura varieties.

desk verdict A useful, carefully written survey of unlikely intersections for Shimura varieties, but the state-of-the-art citations—especially the misdirected Esnault–Groechenig reference—need fixing before this can be trusted as a roadmap. read the letter →

arxiv 2506.02900 v1 pith:LH5CVDYC submitted 2025-06-03 math.NT math.AGmath.LO

Christopher Daw This is my paper · ORCID

classification math.NTmath.AGmath.LO MSC 11G1814G3511G5003C64

keywords unlikelyintersectionsShimuravarietiesAndré–OortconjectureAndré–Pink–ZannierZilber–PinkPila–Zannierstrategyo-minimalityspecialsubvarieties

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 survey gives a short, self-contained introduction to problems of unlikely intersections for pure Shimura varieties—arithmetic quotients of hermitian symmetric domains that parametrise families of abelian varieties and their generalisations—and records where the field currently stands. The organising principle is that a randomly chosen subvariety should meet a distinguished subvariety of larger codimension only rarely, so the union of all such intersections should be sparse unless the subvariety is trapped inside a proper special subvariety. On this basis the survey presents the Zilber–Pink conjecture as the central open problem, with André–Oort and André–Pink–Zannier as its two main corollaries. It reports that André–Oort is now a theorem in full generality and that André–Pink–Zannier has been proved for Shimura varieties of abelian type, and it explains the geometric and arithmetic tools behind these results. A sympathetic reader can use the survey as a reliable roadmap to the definitions, history, strategies, and open arithmetic questions of the field.

What carries the argument

The central mechanism is the Pila–Zannier strategy, a two-part proof scheme. Over a connected component, the uniformisation map from the hermitian symmetric domain to the Shimura variety is definable in the o-minimal structure $\mathbb{R}_{\mathrm{an,exp}}$ on fundamental sets, so the Pila–Wilkie theorem counts algebraic points of bounded height on the preimage of an algebraic subvariety; the geometric theorems of Ax–Lindemann and Ax–Schanuel then force the algebraic points that the count leaves over to lie on weakly special subvarieties. The arithmetic side supplies two inputs: lower bounds for Galois orbits of special points or optimal points, and upper bounds for heights of pre-special points or parameters for totally geodesic subvarieties. A supporting organisational device is Pink's defect $\delta(V) = \dim\langle V\rangle - \dim V$, which measures how far a subvariety is from being special and appears in the optimal-subvariety formulation of Zilber–Pink.

What would settle it

Check the publication status of the two load-bearing citations, [124, Th. 1.1] and [139]. If the André–Oort proof attributed to [124] has a gap that is not repaired, or if the announced large Galois orbits for special points in all Shimura varieties fail to materialise, the survey's claim that André–Oort is fully proved collapses. Similarly, if [139] is withdrawn or restricted to a narrower class than 'abelian type', the survey's André–Pink–Zannier assertion would need revision.

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Extended reading notes

Core claim

The paper's central claim is that the field now possesses a complete solution to its oldest flagship problem, André–Oort, and a structured account of everything else in terms of the Zilber–Pink conjecture. In the survey's telling, André–Oort is a theorem: any irreducible subvariety of a Shimura variety that contains a Zariski-dense set of special points—points whose Mumford–Tate group is a torus—must itself be a special subvariety, as established in [124, Th. 1.1]. It also reports that André–Pink–Zannier, the analogous statement for points lying in a single Hecke orbit (the isogeny class when the Shimura variety is a moduli space of abelian varieties), has been proved for Shimura varieties of abelian type in [139], in the stronger form of generalised Hecke orbits. The surrounding text presents the two proof strategies—the Edixhoven–Klingler–Ullmo–Yafaev route via equidistribution and the Pila–Zannier route via o-minimality and point counting—and isolates large Galois orbits and parameter height bounds as the arithmetic ingredients that still control progress on Zilber–Pink.

Load-bearing premise

The survey's state-of-the-art picture rests on the correctness and current status of results that are cited as preprints or announcements, chiefly the full André–Oort proof and the abelian-type André–Pink–Zannier proof; if those proofs are withdrawn, revised, or misdescribed, the roadmap would mislead.

Editorial extensions

If this is right

If the survey's report is correct, the André–Oort conjecture is settled for every pure Shimura variety: Zariski-density of special points forces a subvariety to be special, closing the Manin–Mumford-type question in this setting.
André–Pink–Zannier for Shimura varieties of abelian type means that a subvariety meeting one Hecke orbit in Zariski-dense fashion must be weakly special, extending the older curve cases to all dimensions in that class.
Because geometric Zilber–Pink is proved in broad generality, the full Zilber–Pink conjecture over number fields is equivalent to an arithmetic statement: finiteness of optimal points on any subvariety defined over $\mathbb{Q}$.
The outstanding bottleneck is a lower bound for Galois orbits of optimal points; currently such bounds are known only in special cases, including curves and certain PEL-type loci in $\mathcal{A}_g$.
Effective methods are beginning to enter: effective point-counting and G-function techniques already yield effective André–Oort results for modular curves and Hilbert modular surfaces, so the survey's roadmap points toward further effective theorems.

Reading between the lines

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

We infer that the survey's structure points to the large Galois orbits estimate for optimal points as the single most productive target: it is the only stated gap between geometric Zilber–Pink and the full conjecture.
We infer that the abelian-type proof of André–Pink–Zannier should transfer, at least in part, to mixed Shimura varieties of Kuga type, since the survey records that the geometric input (Ax–Schanuel) is already available there.
We infer that the quantitative reduction theory of Section 8 is a template for the remaining PEL-type cases (Albert types III and IV), which the survey says are being treated in forthcoming work; if those bounds appear, the corresponding cases of Zilber–Pink follow.
We infer that a fair test of the survey's roadmap is to see whether any new Zilber–Pink case over $\mathbb{Q}$ can be obtained without a new Galois-orbit lower bound; the survey's equivalences suggest it cannot.

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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

This is a good survey of unlikely intersections for pure Shimura varieties, aimed at PhD students and nearby experts. It is a review, so there are no new theorems, but it does something genuinely useful: it compresses the background (Shimura data, special subvarieties, o-minimality, Pila–Wilkie) and lays out the two main proof strategies—Edixhoven–Klingler–Ullmo–Yafaev and Pila–Zannier—with the arithmetic ingredients cleanly separated. The history section is actually helpful, and the references are extensive. I checked the mathematical content in Sections 2 and 3 against my own knowledge; it is standard and accurate, and the descriptions of the strategies match the published literature.

The soft spots are real but mostly fixable. The biggest is the handling of recent preprints. Section 3.2 states André–Oort as a theorem (Thm 3.2.1) citing [124, Th. 1.1], a preprint by Pila–Shankar–Tsimerman. Then Section 6.1.1 says large Galois orbits 'have now been announced in full generality by Pila–Shankar–Tsimerman using new results of Esnault–Groechenig [124]'—but [124] is the Pila–Shankar–Tsimerman preprint, not an Esnault–Groechenig paper, and there is no separate Esnault–Groechenig entry in the bibliography. That is a misdirected reference, and it matters because the survey's promise is to be a faithful map of the current state of the art. The same issue appears, less severely, in the discussion of [139] and in the use of Cassani's thesis [41] in Section 6.2.3. A reader who follows the citations cannot locate the crucial new input, and the text does not consistently distinguish peer-reviewed theorems from announced preprints.

There is also a minor typo ('for the definiton' in 2.7.1). The mathematics itself seems sound; I saw no internal errors.

Who is this for? Early-career researchers and experts in nearby fields. Once the preprint-status labeling and the [124] reference are corrected, this will be a solid entry point. It deserves a serious referee; I would accept with minor revision, but the author should fix the misdirected citation and add clear status cues for announced results before it goes out.

Referee Report

2 major / 5 minor

Summary. The paper is a survey of unlikely intersections for pure Shimura varieties, aimed at early-career researchers and nearby experts. It covers the necessary background on Shimura data and Shimura varieties, special and weakly special subvarieties, and the main conjectures (Zilber–Pink, André–Oort, André–Pink–Zannier, and Mordell–Lang for Shimura varieties). It then reviews the history and the two principal strategies: the Edixhoven–Klingler–Ullmo–Yafaev approach and the Pila–Zannier approach, devoting separate sections to large Galois orbits via G-functions and to parameter height bounds via quantitative reduction theory. The survey does not prove new theorems; it aims to give a concise and reliable map of the current state of the art, including recent preprints and announcements.

Significance. If its reports of the literature are accurate, the survey fills a genuine niche: it is more concise than Pila’s book, more up-to-date than earlier surveys, and specifically focused on pure Shimura varieties while indicating broader settings. Its strengths are the careful definitions in Sections 2 and 3, the clean separation of geometric and arithmetic ingredients in the Pila–Zannier strategy, the explicit statements of the main conjectures, and the inclusion of very recent developments. The survey gives credit to the primary literature and explains the shape of the main strategies without excessive technical detail, which suits the intended audience. The main risk lies in the accuracy and status of citations for a few key recent results, particularly the André–Oort theorem and the large Galois orbits input; these are load-bearing for the survey’s stated goal of being a faithful state-of-the-art roadmap.

major comments (2)

[§6.1.1 and bibliography [124]] In §6.1.1 the sentence “Large Galois orbits for special points have now been announced in full generality by Pila–Shankar–Tsimerman using new results of Esnault–Groechenig [124]” attributes the Esnault–Groechenig input to reference [124], but [124] is the Pila–Shankar–Tsimerman preprint “Canonical Heights on Shimura Varieties and the André–Oort Conjecture”; no separate Esnault–Groechenig entry appears in the bibliography, so a reader following the references cannot locate the crucial input. Moreover, Theorem 3.2.1 in §3.2 and the historical account in §4.1 present André–Oort as an accomplished theorem on the authority of [124, Th. 1.1] without flagging that this proof is currently a preprint or announcement, whereas §6.1.1 itself says the key Galois-orbit ingredient has only been “announced”. Please standardize the status labels (e.g., “announced preprint”, “to appear”, “published”) and supply the correct Esnault–Groechenig reference or delete the attribution.
[§6.2.3] The contradiction step in the Pila–Zannier argument for Arithmetic Zilber–Pink is attributed to Cassani’s PhD thesis [41], which is unpublished. Since this step is essential to the survey’s description of the current status of the Zilber–Pink strategy, the thesis should be explicitly labeled as such, and any published or preprint version should be cited if available.

minor comments (5)

[§2.7.1] There is a typo in “for the definiton of a Siegel set”; it should read “definition”.
[§2.9] The word “homomorhism” in “the G(R)U(C)-conjugacy class of a homomorhism SC → GC” should be “homomorphism”.
[§6.1.1] The phrase “using the the Colmez conjecture on average” contains a duplicated “the”.
[§4.1] “Hilber modular surfaces” should be “Hilbert modular surfaces”.
[Reference list [54] and [55]] References [54] and [55] both list arXiv:2306.13463v1; please clarify whether these are two distinct articles, two versions of the same article, or a single article cited twice, and adjust the entries accordingly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: this expository survey reports external results, and its self-citations are bibliographic rather than load-bearing; a citation inconsistency in Section 6.1.1 affects reliability but not circularity.

full rationale

This is a survey with no original derivation or prediction; its content is a sequence of definitions, statements, and attributions to the literature. I checked for the seven circularity patterns. No step exhibits a target quantity defined in terms of the claimed output, a fitted parameter renamed as prediction, or a uniqueness or ansatz smuggled in through self-citation. The self-citations (e.g., [49], [50], [56], [57], [58], [60], and [5]) function as reports of the author's prior published results, not as premises that make the survey's claims true by definition; a reader can verify each cited result independently. The Andre-Oort statement in Theorem 3.2.1 is attributed to [124, Th. 1.1], not derived here, and the Pila-Zannier sketch uses external ingredients such as Ax-Schanuel [104], Pila-Wilkie [128], height bounds [56], Galois-orbit results [27, 124], and Cassani's thesis [41]; none of these is shown to reduce to an input of this paper. I also weighed the manuscript's own limitations: Section 6.1.1 states that large Galois orbits have been 'announced' using 'new results of Esnault-Groechenig [124]', yet the bibliography entry [124] is the Pila-Shankar-Tsimerman preprint, with no separate Esnault-Groechenig item, and Sections 3.2 and 4.1 present [124] as a theorem rather than an announcement. This is a citation-accuracy and state-of-the-art-reliability defect, not a circularity defect: the survey is not deriving [124] from itself. Accordingly, the circularity score is 0.

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

No original derivation, no fitted parameters, and no new entities appear. The survey's claims are citations to the literature; the only substantive dependence is the accuracy of those citations, which the reader must verify against primary sources.

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Cite this review

Pith. "Pith review of Unlikely intersections in Shimura varieties and beyond: a survey." pith.science (2026). https://pith.science/paper/LH5CVDYC

@misc{pith2026250602900,
  author       = {Pith},
  title        = {Pith review of: Unlikely intersections in Shimura varieties and beyond: a survey},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LH5CVDYC}},
  note         = {Machine review of arXiv:2506.02900}
}

read the original abstract

The aim of this note is to provide a concise introduction to so-called problems of unlikely intersections for (pure) Shimura varieties and to review the current state-of-the-art. In the process, we will touch upon more general settings and some of the results in those contexts.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

165 extracted references · 70 canonical work pages

[124]

J. Pila, A. Shankar, and J Tsimerman. Canonical Heights on Shimura Varieties and the Andr´ e–Oort Conjecture. preprint, available at arXiv:2109.08788

work page Pith review arXiv
[139]

Richard and A

R. Richard and A. Yafaev. Generalised andr´ e-pink-zannier conjecture for shimura varieties of abelian type. Publications math´ ematiques de l’IH´ES, 2025

work page 2025
[41]

O. Cassani. On the Zilber-Pink conjecture via the Large Galois Orbits conjecture . PhD thesis, University of Oxford, 2024

2024
[1]

Andr´ e.G-functions and geometry

Y. Andr´ e.G-functions and geometry . Aspects of Mathematics, E13. Friedr. Vieweg & Sohn, Braun- schweig, 1989

1989
[2]

Y. Andr´ e. Th´ eorie des motifs et interpr´ etation g´ eom´ etrique des valeursp-adiques de G-functions (une introduction). In Number theory (Paris, 1992–1993), volume 215 of London Math. Soc. Lecture Note Ser., pages 37–60. Cambridge Univ. Press, Cambridge, 1995

1992
[3]

Y. Andr´ e. Finitude des couples d’invariants modulaires singuliers sur une courbe alg´ ebrique plane non modulaire. J. Reine Angew. Math. , 505:203–208, 1998

1998
[4]

Andreatta, E.Z

F. Andreatta, E.Z. Goren, B. Howard, and K. Madapusi Pera. Faltings heights of abelian varieties with complex multiplication. Ann. of Math. , 187(2):391–531, 2018

2018
[5]

Aslanyan and C

V. Aslanyan and C. Daw. A note on unlikely intersections in Shimura varieties. J. Number Theory, 260:212–222, 2024

2024

Show all 165 references

[6]

J. Ax. On Schanuel’s conjectures. Ann. of Math. (2) , 93:252–268, 1971

1971
[7]

W. L. Baily, Jr. and A. Borel. Compactification of arithmetic quotients of bounded symmetric domains. Ann. of Math. (2) , 84:442–528, 1966

1966
[8]

Bakker, Y

B. Bakker, Y. Brunebarbe, B. Klingler, and J. Tsimerman. Definability of mixed period maps. J. Eur. Math. Soc. (JEMS) , 26(6):2191–2209, 2024

2024
[9]

Bakker, B

B. Bakker, B. Klingler, and J. Tsimerman. Tame topology of arithmetic quotients and algebraicity of Hodge loci. J. Amer. Math. Soc. , 33(4):917–939, 2020

2020
[10]

Bakker and J

B. Bakker and J. Tsimerman. The Ax-Schanuel conjecture for variations of Hodge structures.Invent. Math., 217(1):77–94, 2019

2019
[11]

G. Baldi. Hodge theory and o-minimality at CIRM. preprint, available at arXiv:2502.03071

arXiv
[12]

G. Baldi. Lieu de Hodge et applications : quelques r´ esultats et conjectures. Habilitation ` a diriger des researches, Sorbonne universit´ e, 2025

2025
[13]

Baldi, B

G. Baldi, B. Klingler, and E. Ullmo. On the distribution of the Hodge locus. Invent. Math. , 235(2):441–487, 2024

2024
[14]

Baldi, R

G. Baldi, R. Richard, and E. Ullmo. Manin–Mumford in arithmetic pencils. preprint, available at arXiv:2105.12027

arXiv
[15]

Baldi and D

G. Baldi and D. Urbanik. Effective atypical intersections and applications to orbit closures. preprint, available at arXiv:2406.16628

arXiv
[16]

Barroero and L

F. Barroero and L. Capuano. Unlikely intersections in families of abelian varieties and the polynomial Pell equation. Proc. Lond. Math. Soc. (3) , 120(2):192–219, 2020

2020
[17]

Barroero and G

F. Barroero and G. Dill. On the Zilber-Pink conjecture for complex abelian varieties. Ann. Sci. ´Ec. Norm. Sup´ er. (4), 55(1):261–282, 2022

2022
[18]

Barroero and G.A

F. Barroero and G.A. Dill. Hecke orbits and the Mordell-Lang conjecture in distinguished categories. preprint, available at arXiv:2401.11193

arXiv
[19]

Barroero and G.A

F. Barroero and G.A. Dill. Distinguished categories and the Zilber–Pink conjecture. to appear in Amer. J. Math., available at https://arxiv.org/abs/2103.07422, 2021. UNLIKELY INTERSECTIONS IN SHIMURA V ARIETIES 25

2021 arXiv
[20]

Barroero, L

F. Barroero, L. K¨ uhne, and H. Schmidt. Unlikely intersections of curves with algebraic subgroups in semiabelian varieties. Selecta Math. (N.S.) , 29(2):Paper No. 18, 37, 2023

2023
[21]

Bertrand

D. Bertrand. Special points and Poincar´ e bi-extensions. with an appendix by Bas Edixhoven, preprint available at arXiv:1104.5178

arXiv
[22]

Bertrand, D

D. Bertrand, D. Masser, A. Pillay, and U. Zannier. Relative Manin-Mumford for semi-Abelian surfaces. Proc. Edinb. Math. Soc. (2) , 59(4):837–875, 2016

2016
[23]

Binyamini

G. Binyamini. Log-Noetherian functions. preprint, available at arXiv:2405.16963

arXiv
[24]

Binyamini

G. Binyamini. Some effective estimates for Andr´ e-Oort inY (1)n. J. Reine Angew. Math., 767:17–35,
[25]

Binyamini and C

G. Binyamini and C. Daw. Effective computations for weakly optimal subvarieties. J. Eur. Math. Soc. (JEMS), 27(5):2155–2186, 2025

2025
[26]

Binyamini and D

G. Binyamini and D. Masser. Effective Andr´ e–Oort for non-compact curves in Hilbert modular varieties. Comptes Rendus. Math´ ematique, 359(3):313–321, 2021

2021
[27]

Binyamini, H

G. Binyamini, H. Schmidt, and A. Yafaev. Lower bounds for Galois orbits of special points on Shimura varieties: a point-counting approach. to appear in Math. Ann. , available at arXiv:2104.05842

arXiv
[28]

Bl´ azquez-Sanz, G

D. Bl´ azquez-Sanz, G. Casale, J. Freitag, and J. Nagloo. A differential approach to Ax-Schanuel, I. preprint, available at arXiv:2102.03384

arXiv
[29]

Bombieri

E. Bombieri. On G-functions. In Recent progress in analytic number theory, Vol. 2 (Durham, 1979) , pages 1–67. 1979

1979
[30]

Bombieri, D

E. Bombieri, D. Masser, and U. Zannier. Intersecting a curve with algebraic subgroups of multi- plicative groups. Internat. Math. Res. Notices , (20):1119–1140, 1999

1999
[31]

Bombieri, D

E. Bombieri, D. Masser, and U. Zannier. Anomalous subvarieties—structure theorems and applica- tions. Int. Math. Res. Not. IMRN , (19):Art. ID rnm057, 33, 2007

2007
[32]

Bombieri, D

E. Bombieri, D. Masser, and U. Zannier. On unlikely intersections of complex varieties with tori. Acta Arith., 133(4):309–323, 2008

2008
[33]

Bombieri and J

E. Bombieri and J. Pila. The number of integral points on arcs and ovals. Duke Math. J. , 59(2):337– 357, 1989

1989
[34]

A. Borel. Introduction aux groupes arithm´ etiques. Publications de l’Institut de Math´ ematique de l’Universit´ e de Strasbourg, XV. Actualit´ es Scientifiques et Industrielles, No. 1341. Hermann, Paris, 1969

1969
[35]

A. Borel. Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem. J. Differential Geometry, 6:543–560, 1972

1972
[36]

Borovoi, C

M. Borovoi, C. Daw, and J. Ren. Conjugation of semisimple subgroups over real number fields of bounded degree. to appear in Proc. Amer. Math. Soc., available at arXiv:1802.05894

arXiv
[37]

R. Brauer. On the Zeta-Functions of Algebraic Number Fields. American Journal of Mathematics , 69(2):243–250, 1947

1947
[38]

F. Breuer. Heights of CM points on complex affine curves. Ramanujan J., 5(3):311–317, 2001

2001
[39]

Cadoret and A

A. Cadoret and A. Kret. Galois-generic points on Shimura varieties. Algebra Number Theory , 10(9):1893–1934, 2016

1934
[40]

L. Capuano. An overview on problems of unlikely intersections in families of abelian varieties. Expo. Math., 41(3):603–617, 2023

2023
[42]

Chai and F

C-L. Chai and F. Oort. Abelian varieties isogenous to a Jacobian. Ann. of Math. (2), 176(1):589–635, 2012

2012
[43]

K.C.T. Chiu. Ax-Schanuel for variations of mixed Hodge structures. Math. Ann., 391(2):1681–1710, 2025. 26 CHRISTOPHER DA W

2025
[44]

Clozel, H

L. Clozel, H. Oh, and E. Ullmo. Hecke operators and equidistribution of Hecke points. Invent. Math., 144(2):327–351, 2001

2001
[45]

Clozel and E

L. Clozel and E. Ullmo. ´ equidistribution de sous-vari´ et´ es sp´ eciales.Ann. of Math. (2) , 161(3):1571– 1588, 2005

2005
[46]

Clozel and E

L. Clozel and E. Ullmo. ´ equidistribution ad´ elique des tores et ´ equidistribution des points CM.Doc. Math., pages 233–260, 2006

2006
[47]

Cohen and G

P.B. Cohen and G. W¨ ustholz. Application of the Andr´ e-Oort conjecture to some questions in transcendence. In A panorama of number theory or the view from Baker’s garden (Z¨ urich, 1999) , pages 89–106. Cambridge Univ. Press, Cambridge, 2002

1999
[48]

Corvaja, D

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

2018
[49]

C. Daw. The Andr´ e-Oort conjecture via o-minimality. In O-minimality and diophantine geome- try, volume 421 of London Math. Soc. Lecture Note Ser. , pages 129–158. Cambridge Univ. Press, Cambridge, 2015

2015
[50]

C. Daw. Degrees of strongly special subvarieties and the Andr´ e-Oort conjecture. J. Reine Angew. Math., 721:81–108, 2016

2016
[51]

C. Daw, A. Gorodnik, and E. Ullmo. Convergence of measures on compactifications of locally symmetric spaces. Math. Zeit., 297:1293–1328, 2021

2021
[52]

C. Daw, A. Gorodnik, and E. Ullmo. The space of homogeneous probability measures on Γ\X S max is compact. Math. Ann., 386(1-2):987–1016, 2023. With an appendix by Jialun Li

2023
[53]

C. Daw, A. Javanpeykar, and L. K¨ uhne. Effective estimates for the degrees of maximal special subvarieties. Selecta Math. (N.S.) , 26(1):Paper No. 2, 31, 2020

2020
[55]

Daw and M

C. Daw and M. Orr. Zilber-pink in a product of modular curves assuming multiplicative degeneration. to appear in Duke Math., preprint available at arXiv:2306.13463v1

arXiv
[56]

Daw and M

C. Daw and M. Orr. Heights of pre-special points of Shimura varieties. Math. Ann., 365(3-4):1305– 1357, 2016

2016
[57]

Daw and M

C. Daw and M. Orr. Quantitative reduction theory and unlikely intersections. International Mathe- matics Research Notices, 2022(20):16138–16195, 2021

2022
[58]

Daw and M

C. Daw and M. Orr. Lattices with skew-Hermitian forms over division algebras and unlikely inter- sections. J. Ec. polytech. Math. , 10:1097–1156, 2023

2023
[59]

Daw and M

C. Daw and M. Orr. Unlikely intersections with E× CM curves inA2. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , 22(4):1705–1745, 2021

2021
[60]

Daw and J

C. Daw and J. Ren. Applications of the hyperbolic Ax–Schanuel conjecture. Compos. Math. , 154:1843–1888, 2018

2018
[61]

Daw and A

C. Daw and A. Yafaev. An unconditional proof of the Andr´ e-Oort conjecture for Hilbert modular surfaces. Manuscripta Math., 135(1-2):263–271, 2011

2011
[62]

P. Deligne. Travaux de Shimura. S´ eminaire Bourbaki, 23` eme ann´ ee (1970/71), Exp. No. 389, pages 123–165., 1971

1970
[63]

P. Deligne. Vari´ et´ es de Shimura: interpr´ etation modulaire, et techniques de construction de mod` eles canoniques. In Automorphic forms, representations and L-functions (Part 2) , Proc. Sympos. Pure Math., XXXIII, pages 247–289. AMS, Providence, R.I., 1979

1979
[64]

G.A. Dill. Unlikely intersections between isogeny orbits and curves. preprint, available at arXiv:1801.05701v4

arXiv
[65]

Edixhoven

B. Edixhoven. Special points on the product of two modular curves. Compositio Math., 114(3):315– 328, 1998

1998
[66]

Edixhoven

B. Edixhoven. On the Andr´ e-Oort conjecture for Hilbert modular surfaces. In Moduli of abelian varieties (Texel Island, 1999) , volume 195 of Progr. Math., pages 133–155. Birkh¨ auser, Basel, 2001. UNLIKELY INTERSECTIONS IN SHIMURA V ARIETIES 27

1999
[67]

Edixhoven

B. Edixhoven. Special points on products of modular curves. Duke Math. J. , 126(2):325–348, 2005

2005
[68]

Edixhoven and R

B. Edixhoven and R. Richard. A mod p variant of the Andr´ e-Oort conjecture.Rend. Circ. Mat. Palermo (2), 69(1):151–157, 2020

2020
[69]

Edixhoven and A

B. Edixhoven and A. Yafaev. Subvarieties of Shimura varieties. Ann. of Math. (2) , 157(2):621–645, 2003

2003
[70]

S. J. Edixhoven, B. J. J. Moonen, and F. Oort. Open problems in algebraic geometry. Bull. Sci. Math., 125(1):1–22, 2001

2001
[71]

Z. Gao. About the mixed Andr´ e-Oort conjecture: reduction to a lower bound for the pure case. C. R. Math. Acad. Sci. Paris , 354(7):659–663, 2016

2016
[72]

Z. Gao. A special point problem of Andr´ e-Pink-Zannier in the universal family of Abelian varieties. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , 17(1):231–266, 2017

2017
[73]

Z. Gao. Towards the Andre-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points. J. Reine Angew. Math. , 732:85–146, 2017

2017
[74]

Z. Gao. Mixed Ax-Schanuel for the universal abelian varieties and some applications. Compos. Math., 156(11):2263–2297, 2020

2020
[75]

Gao and P

Z. Gao and P. Habegger. The relative Manin–Mumford conjecture. preprint, available at arXiv:2303.05045

arXiv
[76]

Gao and B

Z. Gao and B. Klingler. The Ax-Schanuel conjecture for variations of mixed Hodge structures. Math. Ann., 388(4):3847–3895, 2024

2024
[77]

Habegger and J

P. Habegger and J. Pila. Some unlikely intersections beyond Andr´ e–Oort.Compos. Math., 148(1):1– 27, 2012

2012
[78]

Habegger and J

P. Habegger and J. Pila. O-minimality and certain atypical intersections. Ann. Sci. ´Ec. Norm. Sup´ er. (4), 49(4):813–858, 2016

2016
[79]

Etats de la Recherche

P. Habegger, G. R´ emond, T. Scanlon, E. Ullmo, and A. Yafaev. Around the Zilber-Pink conjec- ture/Autour de la conjecture de Zilber-Pink , volume 52 of Panoramas et Synth` eses [Panoramas and Syntheses]. Soci´ et´ e Math´ ematique de France, Paris, 2017. Papers corresponding ...

2017
[80]

M. Hindry. Autour d’une conjecture de Serge Lang. Invent. Math., 94(3):575–603, 1988

1988
[81]

Katz and T

N.M. Katz and T. Oda. On the differentiation of de Rham cohomology classes with respect to parameters. J. Math. Kyoto Univ. , 8:199–213, 1968

1968
[82]

Motives and complex multiplication

B. Klingler. Hodge loci and atypical intersections: conjectures. to appear in “Motives and complex multiplication” (Birkhauser), preprint available at arXiv:1711.09387

arXiv
[83]

Klingler

B. Klingler. Hodge theory, between algebraicity and transcendence. InICM—International Congress of Mathematicians. Vol. 3. Sections 1–4 , pages 2250–2284. EMS Press, Berlin, [2023] ©2023

2023
[84]

Klingler and S

B. Klingler and S. Tayou. The Zilber-Pink conjecture for varieties not defined over Q. preprint, available at arXiv:2504.00865

arXiv
[85]

Klingler, E

B. Klingler, E. Ullmo, and A. Yafaev. The hyperbolic Ax-Lindemann-Weierstrass conjecture. Pub- lications math´ ematiques de l’IH´ES, 123(1):333–360, 2016

2016
[86]

Klingler, E

B. Klingler, E. Ullmo, and A. Yafaev. Bi-algebraic geometry and the Andr´ e-Oort conjecture. In Algebraic geometry: Salt Lake City 2015 , volume 97.2 of Proc. Sympos. Pure Math., pages 319–359. Amer. Math. Soc., Providence, RI, 2018

2015
[87]

Klingler and A

B. Klingler and A. Yafaev. The Andr´ e-Oort conjecture.Ann. of Math. (2) , 180(3):867–925, 2014

2014
[88]

L. K¨ uhne. An effective result of Andr´ e–Oort type.Ann. of Math. (2) , 176(1):651–671, 2012

2012
[89]

Lam and A.N

Y.H.J. Lam and A.N. Shankar. A characteristic p analogue of the Andr´ e–Pink–Zannier conjecture. preprint, available at arXiv:2505.12521

arXiv
[90]

Masser and G

D. Masser and G. W¨ ustholz. Isogeny estimates for abelian varieties, and finiteness theorems. Ann. of Math. (2) , 137(3):459–472, 1993

1993
[91]

Masser and U

D. Masser and U. Zannier. Torsion anomalous points and families of elliptic curves. C. R. Math. Acad. Sci. Paris, 346(9-10):491–494, 2008. 28 CHRISTOPHER DA W

2008
[92]

Masser and U

D. Masser and U. Zannier. Torsion anomalous points and families of elliptic curves. Amer. J. Math., 132(6):1677–1691, 2010

2010
[93]

Masser and U

D. Masser and U. Zannier. Torsion points on families of squares of elliptic curves. Math. Ann., 352(2):453–484, 2012

2012
[94]

Masser and U

D. Masser and U. Zannier. Torsion points on families of products of elliptic curves. Adv. Math., 259:116–133, 2014

2014
[95]

Masser and U

D. Masser and U. Zannier. Torsion points on families of simple abelian surfaces and Pell’s equation over polynomial rings. J. Eur. Math. Soc. (JEMS) , 17(9):2379–2416, 2015. With an appendix by E. V. Flynn

2015
[96]

Masser and U

D. Masser and U. Zannier. Abelian varieties isogenous to no Jacobian.Ann. of Math. (2), 191(2):635– 674, 2020

2020
[97]

Masser and U

D. Masser and U. Zannier. Torsion points, Pell’s equation, and integration in elementary terms. Acta Math., 225(2):227–313, 2020

2020
[98]

Masser and G

D.W. Masser and G. W¨ ustholz. Endomorphism estimates for abelian varieties.Math. Z., 215(4):641– 653, 1994

1994
[99]

Masser and G

D.W. Masser and G. W¨ ustholz. Factorization estimates for abelian varieties. Publications Math´ ematiques de l’IH´ES, 81:5–24, 1995

1995
[100]

G. Maurin. Courbes alg´ ebriques et ´ equations multiplicatives.Math. Ann., 341(4):789–824, 2008

2008
[101]

J. S. Milne. The Action of an Automorphism of C On a Shimura Variety and its Special Points , pages 239–265. Birkh¨ auser Boston, Boston, MA, 1983

1983
[102]

J. S. Milne. Introduction to Shimura varieties. InHarmonic analysis, the trace formula, and Shimura varieties, volume 4 of Clay Math. Proc., pages 265–378. Amer. Math. Soc., Providence, RI, 2005

2005
[103]

B. Mishra. Algorithmic algebra. Texts and Monographs in Computer Science. Springer-Verlag, New York, 1993

1993
[104]

N. Mok, J. Pila, and J. Tsimerman. Ax-Schanuel for Shimura varieties.Ann. of Math. (2), 189(3):945– 978, 2019

2019
[105]

B. Moonen. Linearity properties of Shimura varieties. I. J. Algebraic Geom., 7(3):539–567, 1998

1998
[106]

B. Moonen. Linearity properties of shimura varieties, ii. Compositio Mathematica, 114(1):3–35, 1998

1998
[107]

B. Moonen. The coleman–oort conjecture: Reduction to three key cases. Bulletin of the London Mathematical Society, 54(6):2418–2426, 2022

2022
[108]

Discontinuous groups and abelian varieties

D. Mumford. A note of Shimura’s paper “Discontinuous groups and abelian varieties” . Math. Ann., 181:345–351, 1969

1969
[109]

D. Mumford. Abelian varieties, volume 5 of Tata Institute of Fundamental Research Studies in Mathematics. Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008. With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the seco...

1974
[110]

F. Oort. Canonical liftings and dense sets of CM-points. In Arithmetic geometry (Cortona, 1994) , volume XXXVII of Sympos. Math., pages 228–234. Cambridge Univ. Press, Cambridge, 1997

1994
[111]

M. Orr. Families of abelian varieties with many isogenous fibres.J. Reine Angew. Math., 705:211–231, 2015

2015
[112]

M. Orr. Height bounds and the Siegel property. Algebra & Number Theory, 12(2):455–478, 2018

2018
[113]

M. Orr. Galois conjugates of special points and special subvarieties in Shimura varieties. J. Inst. Math. Jussieu, 20(3):1075–1089, 2021

2021
[114]

M. Orr. Unlikely intersections with Hecke translates of a special subvariety. J. Eur. Math. Soc. , 23(1):1–28, 2021

2021
[115]

G. Papas. Height bounds for certain exceptional points in some variations of Hodge structures. preprint, available at arXiv:2201.11240

arXiv
[116]

G. Papas. Zilber-Pink in Y (1)n: Beyond multiplicative degeneration. preprint, available at arXiv:2402.09487. UNLIKELY INTERSECTIONS IN SHIMURA V ARIETIES 29

arXiv
[117]

G. Papas. Some Cases of the Zilber–Pink Conjecture for Curves in Ag. International Mathematics Research Notices, 2024(5):4160–4206, 08 2023

2024
[118]

Peterzil and S

Y. Peterzil and S. Starchenko. Definability of restricted theta functions and families of abelian varieties. Duke Math. J. , 162(4):731–765, 2013

2013
[119]

J. Pila. On the algebraic points of a definable set. Selecta Math. (N.S.) , 15(1):151–170, 2009

2009
[120]

J. Pila. Rational points of definable sets and results of Andr´ e-Oort-Manin-Mumford type.Int. Math. Res. Not. IMRN , (13):2476–2507, 2009

2009
[121]

J. Pila. O-minimality and the Andr´ e-Oort conjecture forCn. Ann. of Math. (2) , 173(3):1779–1840, 2011

2011
[122]

J. Pila. On a modular Fermat equation. Comment. Math. Helv. , 92(1):85–103, 2017

2017
[123]

J. Pila. Point-counting and the Zilber-Pink conjecture , volume 228 of Cambridge Tracts in Mathe- matics. Cambridge University Press, Cambridge, 2022

2022
[125]

Pila and J

J. Pila and J. Tsimerman. The Andr´ e–Oort conjecture for the moduli space of abelian surfaces. Compos. Math., 149(2):204–216, 2013

2013
[126]

Pila and J

J. Pila and J. Tsimerman. Ax-Lindemann for Ag. Ann. of Math. (2) , 179(2):659–681, 2014

2014
[127]

Pila and J

J. Pila and J. Tsimerman. Ax-Schanuel for the j-function. Duke Math. J. , 165(13):2587–2605, 2016

2016
[128]

Pila and A

J. Pila and A. J. Wilkie. The rational points of a definable set. Duke Math. J. , 133(3):591–616, 2006

2006
[129]

Pila and U

J. Pila and U. Zannier. Rational points in periodic analytic sets and the Manin-Mumford conjecture. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. , 19(2):149–162, 2008

2008
[130]

R. Pink. Arithmetical compactification of mixed Shimura varieties , volume 209 of Bonner Mathe- matische Schriften [Bonn Mathematical Publications] . Universit¨ at Bonn, Mathematisches Institut, Bonn, 1990. Dissertation, Rheinische Friedrich-Wilhelms-Universit¨ at Bonn, Bonn, 1989

1990
[131]

R. Pink. A combination of the conjectures of Mordell–Lang and Andr´ e–Oort. InGeometric methods in algebra and number theory , volume 235 of Progr. Math., pages 251–282. Birkh¨ auser Boston, Boston, MA, 2005

2005
[132]

R. Pink. A common generalization of the conjectures of Andr´ e–Oort, Manin–Mumford, and Mordell– Lang. preprint, available at http://www.math.ethz.ch/˜pink/ftp/AOMMML.pdf, 2005

2005
[133]

G. Prasad. Volumes of S-arithmetic quotients of semi-simple groups. Inst. Hautes ´Etudes Sci. Publ. Math., (69):91–117, 1989. With an appendix by Moshe Jarden and the author

1989
[134]

R. Richard. A two-dimensional arithmetic Andr´ e-Oort problem.Bull. Lond. Math. Soc. , 55(3):1459– 1488, 2023

2023
[135]

Richard and E

R. Richard and E. Ullmo. ´ equidistribution de sous-vari´ et´ es faiblement sp´ eciales eto-minimalit´ e: Andr´ e-Oort g´ eom´ etrique.Ann. Inst. Fourier (Grenoble) , 74(6):2667–2721, 2024. with an appendix by Jiaming Chen, Richard and Ullmo

2024
[136]

Richard and A

R. Richard and A. Yafaev. A Common Generalisation of the Andr´ e-Oort and Andr´ e-Pink-Zannier Conjectures. preprint, available at arXiv:2401.03528

arXiv
[137]

Richard and A

R. Richard and A. Yafaev. Heights on ’Hybrid orbits’ in Shimura varieties. preprint, available at arXiv:2406.20013

arXiv
[138]

Richard and A

R. Richard and A. Yafaev. Height functions on hecke orbits and the generalised andr´ e–pink–zannier conjecture. Compositio Mathematica, 2024

2024
[140]

Saettone

F.M. Saettone. Equidistribution of CM points on Shimura curves and ternary theta series. preprint, available at arXiv:2504.12131

arXiv
[141]

Shankar and J

A.N. Shankar and J. Tsimerman. Abelian varieties not isogenous to Jacobians over global fields. preprint, available at arXiv:2105.02998. 30 CHRISTOPHER DA W

arXiv
[142]

Shankar and J

A.N. Shankar and J. Tsimerman. Unlikely intersections in finite characteristic. Forum Math. Sigma, 6:Paper No. e13, 17, 2018

2018
[143]

Tsimerman

J. Tsimerman. Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special points. J. Amer. Math. Soc. , 25(4):1091–1117, 2012

2012
[144]

Tsimerman

J. Tsimerman. The existence of an abelian variety over Q isogenous to no Jacobian. Ann. of Math. (2), 176(1):637–650, 2012

2012
[145]

Tsimerman

J. Tsimerman. The Andr´ e-Oort conjecture forAg. Ann. of Math. (2) , 187(2):379–390, 2018

2018
[146]

E. Ullmo. Points rationnels des vari´ et´ es de shimura.International Mathematics Research Notices, 2004(76):4109–4125, 01 2004

2004
[147]

E. Ullmo. Equidistribution de sous-vari´ et´ es sp´ eciales. II.J. Reine Angew. Math. , 606:193–216, 2007

2007
[148]

E. Ullmo. Applications du th´ eor` eme d’Ax-Lindemann hyperbolique.Compos. Math., 150(2):175–190, 2014

2014
[149]

Ullmo and A

E. Ullmo and A. Yafaev. A characterization of special subvarieties. Mathematika, 57(2):263–273, 2011

2011
[150]

Ullmo and A

E. Ullmo and A. Yafaev. Galois orbits and equidistribution of special subvarieties: towards the Andr´ e-Oort conjecture.Ann. of Math. (2) , 180(3):823–865, 2014

2014
[151]

Ullmo and A

E. Ullmo and A. Yafaev. Hyperbolic Ax-Lindemann theorem in the cocompact case. Duke Math. J. , 163(2):433–463, 2014

2014
[152]

Ullmo and A

E. Ullmo and A. Yafaev. Nombre de classes des tores de multiplication complexe et bornes inf´ erieures pour les orbites galoisiennes de points sp´ eciaux.Bull. Soc. Math. France , 143(1):197–228, 2015

2015
[153]

Ullmo and A

E. Ullmo and A. Yafaev. Algebraic flows on Shimura varieties.Manuscripta Math., 155(3-4):355–367, 2018

2018
[154]

D. Urbanik. Geometric G-functions and Atypicality. to appear in Duke Math., preprint available at arXiv:2301.01857

arXiv
[155]

van den Dries and C

L. van den Dries and C. Miller. On the real exponential field with restricted analytic functions. Israel J. Math. , 85(1-3):19–56, 1994

1994
[156]

Waterhouse.Introduction to affine group schemes, volume 66 ofGraduate Texts in Mathematics

W.C. Waterhouse.Introduction to affine group schemes, volume 66 ofGraduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1979

1979
[157]

J. Wolfart. Werte hypergeometrischer Funktionen. Invent. Math., 92(1):187–216, 1988

1988
[158]

A. Yafaev. Sous-variet´ es des vari´ et´ es de Shimura. PhD thesis, Universit´ e de Rennes, 2000

2000
[159]

A. Yafaev. On a result of Moonen on the moduli space of principally polarised abelian varieties. Compositio Mathematica, 141(5):1103–1108, 09 2005

2005
[160]

A. Yafaev. A conjecture of Yves Andr´ e’s.Duke Math. J. , 132(3):393–407, 2006

2006
[161]

A height bound for abelian schemes with real ×Q2 multiplication

Zachary Youell. A height bound for abelian schemes with real ×Q2 multiplication. Arch. Math. (Basel), 120(4):381–394, 2023

2023
[162]

Yuan and S Zhang

X. Yuan and S Zhang. On the averaged Colmez conjecture. Ann. of Math. (2) , 187(2):533–638, 2018

2018
[163]

A. Yves. G-functions, motives, and unlikely intersections – old and new. preprint, available at arXiv:2501.09867

arXiv
[164]

U. Zannier. Some problems of unlikely intersections in arithmetic and geometry , volume 181 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, 2012. With appendices by David Masser

2012
[165]

B. Zilber. Exponential sums equations and the Schanuel conjecture. J. London Math. Soc. (2) , 65(1):27–44, 2002

2002
[2020]

With an appendix by Emmanuel Kowalski

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [124]

J. Pila, A. Shankar, and J Tsimerman. Canonical Heights on Shimura Varieties and the Andr´ e–Oort Conjecture. preprint, available at arXiv:2109.08788

work page Pith review arXiv

[2] [139]

Richard and A

R. Richard and A. Yafaev. Generalised andr´ e-pink-zannier conjecture for shimura varieties of abelian type. Publications math´ ematiques de l’IH´ES, 2025

work page 2025

[3] [41]

O. Cassani. On the Zilber-Pink conjecture via the Large Galois Orbits conjecture . PhD thesis, University of Oxford, 2024

2024

[4] [1]

Andr´ e.G-functions and geometry

Y. Andr´ e.G-functions and geometry . Aspects of Mathematics, E13. Friedr. Vieweg & Sohn, Braun- schweig, 1989

1989

[5] [2]

Y. Andr´ e. Th´ eorie des motifs et interpr´ etation g´ eom´ etrique des valeursp-adiques de G-functions (une introduction). In Number theory (Paris, 1992–1993), volume 215 of London Math. Soc. Lecture Note Ser., pages 37–60. Cambridge Univ. Press, Cambridge, 1995

1992

[6] [3]

Y. Andr´ e. Finitude des couples d’invariants modulaires singuliers sur une courbe alg´ ebrique plane non modulaire. J. Reine Angew. Math. , 505:203–208, 1998

1998

[7] [4]

Andreatta, E.Z

F. Andreatta, E.Z. Goren, B. Howard, and K. Madapusi Pera. Faltings heights of abelian varieties with complex multiplication. Ann. of Math. , 187(2):391–531, 2018

2018

[8] [5]

Aslanyan and C

V. Aslanyan and C. Daw. A note on unlikely intersections in Shimura varieties. J. Number Theory, 260:212–222, 2024

2024