Torsion-anomalous intersections in algebraic tori: the openness and finiteness conjectures of Bombieri, Masser, and Zannier

In their 2007 IMRN paper 'Anomalous Subvarieties', Bombieri, Masser, and Zannier introduced anomalous and torsion-anomalous subvarieties of a closed irreducible subvariety $X$ of the algebraic torus $\mathbf{G}_m^n$ and posed, alongside a Bounded Height Conjecture for the deprived set $X^{oa}$, two companion conjectures for the torsion-deprived set $X^{ta}$ (the complement in $X$ of the union of all positive-dimensional torsion-anomalous subvarieties): over $\mathbf{C}$, the set $X^{ta}$ should be Zariski-open in $X$, and $X^{ta}$ should meet the union $\mathcal{H}_{n-r-1}$ of algebraic subgroups of codimension at least $r+1$ (where $r=\dim X$) in only finitely many points. The Bounded Height Conjecture is now a theorem: Habegger proved it for tori (2009) and for abelian varieties, and Kühne extended it to all semiabelian varieties (2020); Habegger's corollaries also give the $X^{oa}$-finiteness over $\overline{\mathbf{Q}}$ and Zilber's conjecture for sufficiently generic subvarieties. The torsion pair, however, remains open in general: openness is known only for codimension-2 subvarieties (Bombieri–Masser–Zannier) and related cases in products of CM elliptic curves (the Viada school). The surviving open core — the Torsion Openness Conjecture and the Torsion Finiteness Conjecture for irreducible $X\subset\mathbf{G}_m^n$ over $\mathbf{C}$ — is the subject of this record.

Background

Let $\mathbf{G}_m^n$ denote the $n$-dimensional algebraic torus, the group variety of invertible diagonal matrices; over $\mathbf{C}$ its points form $(\mathbf{C}^*)^n$. Its algebraic subgroups are defined by systems of monomial equations $x_1^{a_1}\cdots x_n^{a_n}=1$, and their identity components are again tori, called subtori. A torsion point is a point of finite order. Torsion points are Zariski-dense in $\mathbf{G}_m^n$ (dense for the Zariski topology, whose closed sets are the algebraic subsets), so a subvariety can meet them, and translates of subgroups, in unexpectedly large sets. If $X\subset\mathbf{G}_m^n$ is closed irreducible of dimension $r$ and $K$ is a translate of an algebraic subgroup, the expected dimension of $X\cap K$ is $r+\dim K-n$; a positive-dimensional irreducible subvariety $Y$ of $X$ lying in such a translate, with $\dim Y>r+\dim K-n$, is called an anomalous subvariety of $X$ (single points are excluded from the notion: every point of $X$ lies in the one-point translate $\{p\}$, which already has excess dimension whenever $r<n$). Bombieri, Masser, and Zannier proved a structure theorem: the union of the maximal anomalous subvarieties of $X$ is Zariski-closed, with finitely many maximal components described inductively, and its complement $X^{oa}$, the deprived set, is Zariski-open in $X$. If the translating point of $K$ is required to be a torsion point, one obtains torsion-anomalous subvarieties of $X$ and the torsion-deprived set $X^{ta}$, the complement in $X$ of the union of all positive-dimensional torsion-anomalous subvarieties of $X$, or equivalently of the maximal ones; since fewer subvarieties are removed than in $X^{oa}$, $X^{oa}\subseteq X^{ta}$ and the inclusion can be strict. For $0\le s\le n-1$ write $\mathcal{H}_s$ for the union of the algebraic subgroups of $\mathbf{G}_m^n$ of dimension at most $s$ (equivalently, of codimension at least $n-s$).

The arithmetic input is the absolute logarithmic Weil height $h$, a nonnegative function on algebraic points measuring their arithmetic complexity, for which Northcott's theorem gives finiteness of height-bounded sets of algebraic points of bounded degree. In 2007 Bombieri, Masser, and Zannier posed the Bounded Height Conjecture: the height is bounded on $X^{oa}\cap\mathcal{H}_{n-r}$, i.e. on the deprived points lying in subgroups of complementary codimension. Their 1999 theorem for curves and the hypersurface case (their 1995 study of algebraic points together with Zannier's 2000 appendix to Schinzel's book on reducibility of polynomials) motivated the conjecture. Independently, Zilber (2002), from a model-theoretic study of the formal theory of exponentiation, conjectured that if $X$ lies in no proper algebraic subgroup then $X(\overline{\mathbf{Q}})\cap\mathcal{H}_{n-r-1}$ is not Zariski-dense in $X$; Pink (2005) proposed a common generalization of the André–Oort, Manin–Mumford, and Mordell–Lang conjectures for mixed Shimura varieties (Shimura varieties are higher-dimensional moduli spaces generalizing modular curves; André–Oort and Manin–Mumford describe special and torsion structure on them). These are the 'unlikely intersections' problems: excess intersection with special structures must be explained by group-theoretic or moduli-theoretic structure. Habegger proved the Bounded Height Conjecture for tori in 2009, in a stronger height-cone form, and as corollaries obtained that $X^{oa}\cap\mathcal{H}_{n-r-1}$ is finite over $\overline{\mathbf{Q}}$ and that Zilber's conjecture holds for sufficiently generic subvarieties defined over $\overline{\mathbf{Q}}$; he also proved the abelian analogue (abelian varieties are projective algebraic groups, for example elliptic curves and Jacobians of curves), and Kühne (2020) established the conjecture for all semiabelian varieties (extensions of abelian varieties by tori). What these theorems cannot reach is the torsion-anomalous part of $X$ over $\mathbf{C}$: with $X^{oa}$ replaced by the larger $X^{ta}$, and over $\mathbf{C}$ where Northcott's theorem is unavailable, the corresponding openness and finiteness statements are conjectures of the same 2007 paper, known so far only in special cases — codimension 2, proved by Bombieri–Masser–Zannier themselves, and products of elliptic curves with complex multiplication (elliptic curves whose endomorphism ring is larger than $\mathbf{Z}$), studied by the Viada school.

Problem Statement

Let $n\ge 2$ and let $X\subset\mathbf{G}_m^n$ be a closed irreducible subvariety of dimension $r$ defined over $\mathbf{C}$. Recall that, in the sense of Bombieri, Masser, and Zannier, a positive-dimensional irreducible subvariety $Y\subseteq X$ is torsion-anomalous if $Y$ is contained in a translate $aH$ of a proper algebraic subgroup $H\subset\mathbf{G}_m^n$ by a torsion point $a$ and $\dim Y>r+\dim H-n$ (single points are excluded from this notion); that $X^{ta}$ is the complement in $X$ of the union of all torsion-anomalous subvarieties of $X$, or equivalently of the maximal ones (maximal with respect to inclusion); and that $\mathcal{H}_s$ is the union of the algebraic subgroups of $\mathbf{G}_m^n$ of dimension at most $s$ (equivalently, of codimension at least $n-s$).

Prove or refute, in this generality:

(1) Torsion Openness Conjecture: $X^{ta}$ is a Zariski-open subset of $X$.

(2) Torsion Finiteness Conjecture: $X^{ta}\cap\mathcal{H}_{n-r-1}$ is a finite set.

References

1. E. Bombieri, D. Masser, U. Zannier, Anomalous Subvarieties—Structure Theorems and Applications, International Mathematics Research Notices 2007 (2007), no. 19, Article ID rnm057, 33 pp., DOI 10.1093/imrn/rnm057, https://doi.org/10.1093/imrn/rnm057
2. E. Bombieri, D. Masser, U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups, International Mathematics Research Notices 1999 (1999), no. 20, pp. 1119–1140, DOI 10.1155/S1073792899000628, https://doi.org/10.1155/S1073792899000628, zbMATH https://zbmath.org/1402564 (Zbl 0938.11031)
3. E. Bombieri, U. Zannier, Algebraic points on subvarieties of G_m^n, International Mathematics Research Notices 1995 (1995), no. 7, pp. 333–347, DOI 10.1155/S1073792895000250, https://doi.org/10.1155/S1073792895000250, zbMATH https://zbmath.org/812655 (Zbl 0848.11030)
4. E. Bombieri, D. Masser, U. Zannier, On unlikely intersections of complex varieties with tori, Acta Arithmetica 133 (2008), no. 4, pp. 309–323, DOI 10.4064/aa133-4-2, https://doi.org/10.4064/aa133-4-2
5. A. Schinzel, Polynomials with Special Regard to Reducibility, with an appendix by Umberto Zannier, Encyclopedia of Mathematics and its Applications 77, Cambridge University Press, Cambridge, 2000, ix+558 pp. (appendix: pp. 517–539), DOI 10.1017/cbo9780511542916, https://doi.org/10.1017/cbo9780511542916, zbMATH Zbl 0956.12001
6. P. Habegger, On the Bounded Height Conjecture, International Mathematics Research Notices 2009 (2009), no. 5, pp. 860–886, DOI 10.1093/imrn/rnn149, https://doi.org/10.1093/imrn/rnn149
7. P. Habegger, Intersecting subvarieties of G_m^n with algebraic subgroups, Mathematische Annalen 342 (2008), no. 2, pp. 449–466, DOI 10.1007/s00208-008-0242-3, https://doi.org/10.1007/s00208-008-0242-3
8. P. Habegger, Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension, Inventiones mathematicae 176 (2009), no. 2, pp. 405–447, DOI 10.1007/s00222-008-0170-6, https://doi.org/10.1007/s00222-008-0170-6
9. G. Maurin, Courbes algébriques et équations multiplicatives, Mathematische Annalen 341 (2008), no. 4, pp. 789–824, DOI 10.1007/S00208-008-0212-9, https://doi.org/10.1007/S00208-008-0212-9
10. L. Kühne, The bounded height conjecture for semiabelian varieties, Compositio Mathematica 156 (2020), no. 7, pp. 1405–1456, DOI 10.1112/S0010437X20007198, https://doi.org/10.1112/S0010437X20007198, arXiv:1703.03891
11. B. Zilber, Exponential sums equations and the Schanuel conjecture, Journal of the London Mathematical Society 65 (2002), no. 1, pp. 27–44, DOI 10.1112/S0024610701002861, https://doi.org/10.1112/S0024610701002861
12. R. Pink, A Common Generalization of the Conjectures of André-Oort, Manin-Mumford, and Mordell-Lang, preprint, April 17, 2005, 13 pp., https://people.math.ethz.ch/~pink/ftp/AOMMML.pdf
13. S. Checcoli, F. Veneziano, E. Viada, On torsion anomalous intersections, Rendiconti Lincei — Matematica e Applicazioni 25 (2014), no. 1, pp. 1–36, DOI 10.4171/rlm/666, https://doi.org/10.4171/rlm/666, arXiv:1204.1435
14. S. Checcoli, E. Viada, On the torsion anomalous conjecture in CM abelian varieties, Pacific Journal of Mathematics 271 (2014), no. 2, pp. 321–345, DOI 10.2140/pjm.2014.271.321, https://doi.org/10.2140/pjm.2014.271.321
15. P. Hubschmid, E. Viada, An addendum to the elliptic torsion anomalous conjecture in codimension 2, Rendiconti del Seminario Matematico della Università di Padova 141 (2019), pp. 209–220, DOI 10.4171/RSMUP/23, https://doi.org/10.4171/RSMUP/23, arXiv:1604.05741
16. D. Ghioca, K. D. Nguyen, Dynamical anomalous subvarieties: structure and bounded height theorems, Advances in Mathematics 288 (2016), pp. 1433–1462, DOI 10.1016/j.aim.2015.11.015, https://doi.org/10.1016/j.aim.2015.11.015, arXiv:1408.5455
17. F. Barroero, L. Kühne, H. Schmidt, Unlikely intersections of curves with algebraic subgroups in semiabelian varieties, Selecta Mathematica (New Series) 29 (2023), no. 2, Paper No. 18, 37 pp., DOI 10.1007/s00029-022-00823-w, https://doi.org/10.1007/s00029-022-00823-w, arXiv:2108.12405
18. M. Orr, Explicit height bounds for G-functions and unlikely intersections with lines in tori, preprint (2026), arXiv:2607.09565, https://arxiv.org/abs/2607.09565
19. L. Capuano, Exploring Zilber's conjecture on atypical intersections in tori and beyond, Journal of the London Mathematical Society 113 (2026), no. 4, DOI 10.1112/jlms.70536, https://doi.org/10.1112/jlms.70536
20. U. Zannier, with an appendix by David Masser, Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181), Annals of Mathematics Studies 181, Princeton University Press, Princeton, 2012, ISBN 9780691153704, DOI 10.23943/princeton/9780691153704.001.0001, https://doi.org/10.23943/princeton/9780691153704.001.0001