HEXAGONbeta

An effective André–Oort theorem for Shimura varieties

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorymath.AG — Algebraic Geometry

Contributed by Manvir Jaswal ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We give effective bounds for the special subvarieties VV that are maximal in a subvariety ZZ of a Shimura variety, in terms of the degree of ZZ: for the degree of VV, with no hypothesis; and for the discriminant of a number field LVL_V attached to VV and the Klingler–Yafaev invariant αV\alpha_V of the level, if for some δ>0\delta>0 the Dedekind zeta function of LVL_V has no zero with real part greater than 1−δ1-\delta. Here ZZ is geometrically irreducible and defined over a number field, the level is neat and of product form, and the bounds are explicit in the degree of ZZ and, for LVL_V and αV\alpha_V, also in the degree of the field and δ\delta, with constants that are effectively computable from the Shimura datum, the level and representatives of the connected components of the adjoint Shimura variety. A preprint of Cureton and Thorner, whose proof has been formalized in Lean, establishes such a half-plane with δ=1/32\delta=1/32 for every number field; granting it, all these bounds hold with no further hypothesis. For abelian type, an algorithm lists these subvarieties from the bounds, given a finite étale cover by a Shimura variety of Hodge type inside one of Mumford's moduli spaces and equations for the preimage of ZZ.

Provenance statement

This paper and its results were produced by an AI research system that the author designed, built and directs, running on Anthropic's Claude Fable 5.2 in Claude Code. The author chooses the problems and prioritizes the work. Separate AI agents checked each result, and a second program confirmed the computations the proofs use, with any exceptions listed in the public deposit. The system also searched the literature and checked the citations against reliable sources. The author read the paper, reran the computations from the deposit and takes full responsibility for all of the paper's contents.

Tools used

Anthropic
Claude FableVersion 5.2
Anthropic
Claude Code
PARI/GP
PARI/GPVersion 2.17.3
Python
PythonVersion 3.12

References

  1. Alexeev, Valery. Log canonical singularities and complete moduli of stable pairs. 1996. Preprint, \arXivalg-geom/9608013
  2. Amoroso, Francesco; Dvornicich, Roberto. Lower bounds for the height and size of the ideal class group in CM-fields. Monatsh. Math., vol. 138, no. 2, pp. 85–94. 2003. \DOI10.1007/s00605-002-0499-7
  3. André, Yves. Mumford–Tate groups of mixed Hodge structures and the theorem of the fixed part. Compositio Math., vol. 82, no. 1, pp. 1–24. 1992
  4. Ash, Avner; Mumford, David; Rapoport, Michael; Tai, Yung-Sheng. Smooth Compactifications of Locally Symmetric Varieties. Cambridge University Press. 2010. \DOI10.1017/CBO9780511674693
  5. AxiomMath. DedekindZetaZeroFree: a Lean formalization of a zero-free half-plane for Dedekind zeta functions. 2026. https://github.com/AxiomMath/DedekindZetaZeroFree, commit 09afbb4clink
  6. Barakat, Mohamed; Lange-Hegermann, Markus. An algorithmic approach to Chevalley's theorem on images of rational morphisms between affine varieties. Math. Comp., vol. 91, no. 333, pp. 451–490. 2022. \DOI10.1090/mcom/3632, \arXiv1911.10411v3
  7. Bilu, Yuri; Masser, David; Zannier, Umberto. An effective ``theorem of André'' for CM-points on a plane curve. Math. Proc. Cambridge Philos. Soc., vol. 154, no. 1, pp. 145–152. 2013
  8. Binyamini, Gal. Some effective estimates for André–Oort in Y(1)nY(1)^n. J. Reine Angew. Math., vol. 767, pp. 17–35. 2020. With an appendix by Emmanuel Kowalski, \DOI10.1515/crelle-2019-0028, \arXiv1809.05302
  9. Binyamini, Gal. Point counting for foliations over number fields. Forum Math. Pi, vol. 10, pp. Paper No. e6. 2022. \DOI10.1017/fmp.2021.20, \arXiv2009.00892
  10. Binyamini, Gal; Daw, Christopher. Effective computations for weakly optimal subvarieties. J. Eur. Math. Soc. (JEMS), vol. 27, no. 5, pp. 2155–2186. 2025. \DOI10.4171/JEMS/1408, \arXiv2105.12760
Show all 87 referencesShow fewer references
  1. Binyamini, Gal; Jones, Gareth O.; Schmidt, Harry; Thomas, Margaret E. M. An effective Pila–Wilkie theorem for sets definable using Pfaffian functions, with some diophantine applications. J. Eur. Math. Soc. (JEMS). 2026. Published online first, \DOI10.4171/JEMS/1761, \arXiv2301.09883v1; theorem numbers refer to v1
  2. Binyamini, Gal; Masser, David. Effective André–Oort for non-compact curves in Hilbert modular varieties. C. R. Math. Acad. Sci. Paris, vol. 359, pp. 313–321. 2021. \arXiv2101.06412
  3. Binyamini, Gal; Schmidt, Harry; Yafaev, Andrei. Lower bounds for Galois orbits of special points on Shimura varieties: a point-counting approach. Math. Ann., vol. 385, no. 1-2, pp. 961–973. 2023. \DOI10.1007/s00208-021-02309-0, \arXiv2104.05842
  4. Bruhat, François; Tits, Jacques. Groupes réductifs sur un corps local. I. Données radicielles valuées. Inst. Hautes Études Sci. Publ. Math., vol. 41, pp. 5–251. 1972. \DOI10.1007/BF02715544
  5. Bruhat, François; Tits, Jacques. Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d'une donnée radicielle valuée. Inst. Hautes Études Sci. Publ. Math., vol. 60, pp. 5–184. 1984. \DOI10.1007/BF02700560
  6. Clozel, Laurent; Ullmo, Emmanuel. Équidistribution de sous-variétés spéciales. Ann. of Math. (2), vol. 161, no. 3, pp. 1571–1588. 2005. \DOI10.4007/annals.2005.161.1571
  7. Conrad, Brian. Reductive group schemes. Autour des schémas en groupes. Vol. I, vol. 42/43, pp. 93–444. 2014
  8. Cureton, Dean; Thorner, Jesse. A quasi-Riemann hypothesis for Dedekind zeta functions. 2026. Preprint, Hexagon 2610.00200v2, https://hexagonmath.org/2610.00200v2hexagon
  9. Das, Sourabhashis; Kadiri, Habiba; Ng, Nathan. An effective version of Chebotarev's density theorem. 2025. Preprint, \arXiv2508.09480v1
  10. Daw, Christopher. Degrees of strongly special subvarieties and the André–Oort conjecture. J. Reine Angew. Math., vol. 721, pp. 81–108. 2016. \DOI10.1515/crelle-2014-0062
  11. Daw, Christopher; Javanpeykar, Ariyan; Kühne, Lars. Effective estimates for the degrees of maximal special subvarieties. Selecta Math. (N.S.), vol. 26, no. 1, pp. Paper No. 2. 2020. \DOI10.1007/s00029-019-0528-1, \arXiv1812.04341v2; page numbers refer to v2
  12. Daw, Christopher; Orr, Martin. Heights of pre-special points of Shimura varieties. Math. Ann., vol. 365, no. 3-4, pp. 1305–1357. 2016. \DOI10.1007/s00208-015-1328-3, \arXiv1502.00822v5; we cite the numbering of the arXiv version
  13. Deligne, Pierre. Travaux de Shimura. Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, vol. 244, pp. 123–165. 1971
  14. Deligne, Pierre. Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. Automorphic Forms, Representations and LL-Functions, Part 2, vol. 33, pp. 247–289. 1979
  15. Di Cerbo, Gabriele; Di Cerbo, Luca F. Effective results for complex hyperbolic manifolds. J. Lond. Math. Soc. (2), vol. 91, no. 1, pp. 89–104. 2015. \DOI10.1112/jlms/jdu065, \arXiv1212.0501v5; corollary numbers refer to v5
  16. Edixhoven, Bas. On the André–Oort conjecture for Hilbert modular surfaces. Moduli of Abelian Varieties (Texel Island, 1999), vol. 195, pp. 133–155. 2001. \arXivmath/9911272
  17. Edixhoven, Bas; Yafaev, Andrei. Subvarieties of Shimura varieties. Ann. of Math. (2), vol. 157, no. 2, pp. 621–645. 2003. \arXivmath/0105241
  18. Faltings, Gerd; Chai, Ching-Li. Degeneration of Abelian Varieties. Springer, vol. 22. 1990
  19. Feng, Ruyong. Hrushovski's algorithm for computing the Galois group of a linear differential equation. Adv. in Appl. Math., vol. 65, pp. 1–37. 2015. \arXiv1312.5029v3
  20. Fowler, Guy. André's theorem and weakly bounded height. 2026. Preprint, \arXiv2606.29369
  21. Fowler, Guy. Some uniform effective results on André–Oort for sums of powers in Cn\mathbb{C}^n. Math. Proc. Cambridge Philos. Soc., vol. 180, no. 3, pp. 607–641. 2026. \DOI10.1017/S0305004125101825, \arXiv2405.06456
  22. Fowler, Guy. A uniform effective André–Oort result. 2026. Preprint, \arXiv2609.18934
  23. Frei, Christopher; Sofos, Efthymios; Widmer, Martin. Some applications of the quasi-Riemann hypothesis to class groups and small generators of number fields. 2026. Preprint, \arXiv2610.18491v2
  24. Fujino, Osamu. Effective base point free theorem for log canonical pairs II. Angehrn–Siu type theorems. Michigan Math. J., vol. 59, no. 2, pp. 303–312. 2010. \arXiv0709.0336
  25. Fulton, William. Intersection Theory. Springer, vol. 2. 1998
  26. Fulton, William; Lazarsfeld, Robert. Positivity and excess intersection. Enumerative Geometry and Classical Algebraic Geometry (Nice, 1981), vol. 24, pp. 97–105. 1982
  27. Gille, Philippe. Le problème de Kneser–Tits. Séminaire Bourbaki, Vol. 2007/2008, Exp. No. 983, vol. 326, pp. 39–81. 2009
  28. Greuel, Gert-Martin; Laplagne, Santiago; Seelisch, Frank. Normalization of rings. J. Symbolic Comput., vol. 45, pp. 887–901. 2010. \arXiv0904.3561v3
  29. Grothendieck, Alexander. Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math., vol. 8, pp. 5–222. 1961
  30. Grothendieck, Alexander. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. IV. Inst. Hautes Études Sci. Publ. Math., vol. 32, pp. 5–361. 1967
  31. Grothendieck, Alexander. Revêtements étales et groupe fondamental (SGA 1). Soc. Math. France, vol. 3. 2003. Séminaire de géométrie algébrique du Bois Marie 1960–61, augmented by two exposés of M. Raynaud; updated reprint of the 1971 original, \arXivmath/0206203
  32. Hamm, Helmut A.; Lê Dũng Tráng. Lefschetz theorems on quasi-projective varieties. Bull. Soc. Math. France, vol. 113, pp. 123–142. 1985
  33. Hartshorne, Robin. Algebraic Geometry. Springer, vol. 52. 1977
  34. Hu, Boyang. A seven-eighths zero-free half-plane for unitary Weil LL-functions. 2026. Preprint, Hexagon 2610.00216v2, https://hexagonmath.org/2610.00216v2; Lean formalization https://github.com/Rogerhu12/hecke-seven-eighths, commit c77efa92hexagon
  35. Hulek, Klaus; Sankaran, Gregory K. The geometry of Siegel modular varieties. Higher Dimensional Birational Geometry (Kyoto, 1997), vol. 35, pp. 89–156. 2002. \arXivmath/9810153
  36. Jaswal, Manvir. Programs and data for ``An effective André–Oort theorem for Shimura varieties''. 2026. Zenodo, \DOI10.5281/zenodo.24136430
  37. Katz, Nicholas M. Nilpotent connections and the monodromy theorem: applications of a result of Turrittin. Inst. Hautes Études Sci. Publ. Math., vol. 39, pp. 175–232. 1970
  38. Klingler, Bruno; Yafaev, Andrei. The André–Oort conjecture. Ann. of Math. (2), vol. 180, no. 3, pp. 867–925. 2014. \DOI10.4007/annals.2014.180.3.2, \arXiv1209.0936v2; page numbers refer to v2
  39. Kollár, János. Singularities of pairs. Algebraic Geometry–Santa Cruz 1995, vol. 62, Part 1, pp. 221–287. 1997
  40. Kühne, Lars. An effective result of André–Oort type. Ann. of Math. (2), vol. 176, no. 1, pp. 651–671. 2012
  41. Lagarias, Jeffrey C.; Odlyzko, Andrew M. Effective versions of the Chebotarev density theorem. Algebraic Number Fields: LL-Functions and Galois Properties (Durham, 1975), pp. 409–464. 1977
  42. Lella, Paolo. Computable Hilbert schemes. 2012. PhD thesis, Università di Torino; \arXiv1202.4384v1
  43. Malle, Gunter; Testerman, Donna. Linear Algebraic Groups and Finite Groups of Lie Type. Cambridge University Press, vol. 133. 2011
  44. Milne, James S. Introduction to Shimura varieties. Harmonic Analysis, the Trace Formula, and Shimura Varieties, vol. 4. 2005. pp. 265–378; revised version (2017) at https://www.jmilne.org/math/xnotes/svi.pdf, whose display, footnote and page numbers we citelink
  45. Milne, James S. Abelian Varieties. 2008. Course notes, v2.00, https://www.jmilne.org/math/link
  46. Milne, James S. Algebraic Number Theory. 2020. Course notes, v3.08, https://www.jmilne.org/math/link
  47. Milne, James S. Class Field Theory. 2020. Course notes, v4.03, https://www.jmilne.org/math/link
  48. Mumford, David. On the equations defining abelian varieties. I. Invent. Math., vol. 1, pp. 287–354. 1966
  49. Mumford, David. On the equations defining abelian varieties. II. Invent. Math., vol. 3, pp. 75–135. 1967
  50. Mumford, David. Hirzebruch's proportionality theorem in the non-compact case. Invent. Math., vol. 42, pp. 239–272. 1977
  51. OpenAI. Primitive roots for every admissible integer base. 2026. Preprint, October 4, 2026, https://github.com/openai/mathlink
  52. OpenAI. The quasi-Riemann hypothesis: a zero-free half-plane ℜ(s)>7/8\Re(s)>7/8. 2026. Preprint, September 30, 2026, https://github.com/openai/mathlink
  53. Papas, Georgios. Effective Brauer–Siegel on some curves in Y(1)nY(1)^n. Math. Ann., vol. 394, pp. Paper No. 84. 2026. \arXiv2310.04943
  54. Pila, Jonathan; Shankar, Ananth N.; Tsimerman, Jacob. Canonical heights on Shimura varieties and the André–Oort conjecture. 2021. With an appendix by Hélène Esnault and Michael Groechenig. Preprint, \arXiv2109.08788v4
  55. Pila, Jonathan; Wilkie, Alex J. The rational points of a definable set. Duke Math. J., vol. 133, no. 3, pp. 591–616. 2006
  56. Pila, Jonathan; Zannier, Umberto. Rational points in periodic analytic sets and the Manin–Mumford conjecture. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. (9), vol. 19, no. 2, pp. 149–162. 2008
  57. Pink, Richard. Arithmetical Compactification of Mixed Shimura Varieties. Universität Bonn, Mathematisches Institut, vol. 209. 1990. Dissertation, Bonn 1989
  58. Platonov, Vladimir P. The problem of strong approximation and the Kneser–Tits conjecture for algebraic groups. Izv. Akad. Nauk SSSR Ser. Mat., vol. 33, no. 6, pp. 1211–1219. 1969. English translation in Math. USSR-Izv. 3 (1969), 1139–1147, \DOI10.1070/im1969v003n06abeh000838
  59. Platonov, Vladimir P. Addendum to the paper ``The problem of strong approximation and the Kneser–Tits conjecture for algebraic groups''. Izv. Akad. Nauk SSSR Ser. Mat., vol. 34, no. 4, pp. 775–777. 1970. English translation in Math. USSR-Izv. 4 (1970), 784–786, \DOI10.1070/IM1970v004n04ABEH000932
  60. Polishchuk, Alexander. Determinant bundles for abelian schemes. Compositio Math., vol. 121, no. 3, pp. 221–245. 2000. \DOI10.1023/A:1001896315311, \arXivalg-geom/9703021v2
  61. Prasad, Gopal. Elementary proof of a theorem of Bruhat–Tits–Rousseau and of a theorem of Tits. Bull. Soc. Math. France, vol. 110, pp. 197–202. 1982. \DOI10.24033/bsmf.1959
  62. Prasad, Gopal. Volumes of SS-arithmetic quotients of semi-simple groups. Inst. Hautes Études Sci. Publ. Math., vol. 69, pp. 91–114. 1989. \DOI10.1007/BF02698841
  63. Rosser, J. Barkley; Schoenfeld, Lowell. Approximate formulas for some functions of prime numbers. Illinois J. Math., vol. 6, pp. 64–94. 1962
  64. Scanlon, Thomas. Automatic uniformity. Int. Math. Res. Not., no. 62, pp. 3317–3326. 2004. \DOI10.1155/S1073792804140816
  65. Singer, Michael F. Galois theory of linear differential equations. Algebraic Theory of Differential Equations, vol. 357, pp. 1–82. 2009. \DOI10.1017/CBO9780511721564.002, \arXiv0712.4124v2
  66. Stark, Harold M. Some effective cases of the Brauer–Siegel theorem. Invent. Math., vol. 23, pp. 135–152. 1974
  67. Taylor, Donald E. The Geometry of the Classical Groups. Heldermann, vol. 9. 1992
  68. The Stacks project authors. The Stacks project. 2026. https://stacks.math.columbia.edulink
  69. Tits, Jacques. Algebraic and abstract simple groups. Ann. of Math. (2), vol. 80, no. 2, pp. 313–329. 1964. \DOI10.2307/1970394
  70. Tits, Jacques. Reductive groups over local fields. Automorphic Forms, Representations and LL-Functions, Part 1, vol. 33, pp. 29–69. 1979
  71. Tsimerman, Jacob. Brauer–Siegel for arithmetic tori and lower bounds for Galois orbits of special points. J. Amer. Math. Soc., vol. 25, no. 4, pp. 1091–1117. 2012. \DOI10.1090/S0894-0347-2012-00739-5, \arXiv1103.5619v3; we cite the numbering of the arXiv version
  72. Tsimerman, Jacob. The André–Oort conjecture for Ag\mathcal{A}_g. Ann. of Math. (2), vol. 187, no. 2, pp. 379–390. 2018. \arXiv1506.01466v5; corollary numbers refer to v5
  73. Ullmo, Emmanuel. Équidistribution de sous-variétés spéciales. II. J. Reine Angew. Math., vol. 606, pp. 193–216. 2007
  74. Ullmo, Emmanuel; Yafaev, Andrei. Galois orbits and equidistribution of special subvarieties: towards the André–Oort conjecture. Ann. of Math. (2), vol. 180, no. 3, pp. 823–865. 2014. \DOI10.4007/annals.2014.180.3.1, \arXiv1209.0934v2; page numbers refer to v2
  75. Ullmo, Emmanuel; Yafaev, Andrei. Nombre de classes des tores de multiplication complexe et bornes inférieures pour les orbites galoisiennes de points spéciaux. Bull. Soc. Math. France, vol. 143, no. 1, pp. 197–228. 2015. \arXiv1209.0942v2; page numbers refer to v2
  76. Urbanik, David. Sets of special subvarieties of bounded degree. Compositio Math., vol. 159, no. 3, pp. 616–657. 2023. \DOI10.1112/S0010437X23007029, \arXiv2109.07663v1; we cite the numbering of v1
  77. Yafaev, Andrei. A conjecture of Yves André's. Duke Math. J., vol. 132, no. 3, pp. 393–407. 2006. \arXivmath/0302125v1; we cite the numbering of the arXiv version

Version history

  1. v1 · 73 pagesSubmitted by Manvir JaswalInitial depositCurrentNov 11, 2026