Let Y be an irreducible subvariety of bf G_m^n defined over bf C. For algebraic subgroups H⊂ bf G_m^n satisfying dim Y+dim H≤ n-1, consider the union of the intersections Ycap H. Zilber's conjecture on intersections with tori. If bigcup_substackH⊂ bf G_m^n dim Y+dim H≤ n-1Ycap H is Zariski dense in Y, then Y is contained in a proper algebraic subgroup of bf G_m^n. Zilber's conjecture is stated more generally for semi-abelian varieties, and the conjecture above is open, although many partial results are known.

Zilber's conjecture on intersections with tori

About 12 years old · traced to 2014

Let $\mathcal{Y}$ be an irreducible subvariety of ${\bf G}_m^n$ defined over ${\bf C}$ . For algebraic subgroups $H\subset {\bf G}_m^n$ satisfying

\dim \mathcal{Y}+\dim H\le n-1,

consider the union of the intersections $\mathcal{Y}\cap H$ . Zilber's conjecture on intersections with tori. If

\bigcup_{\substack{H\subset {\bf G}_m^n\\ \dim \mathcal{Y}+\dim H\le n-1}}\mathcal{Y}\cap H

is Zariski dense in $\mathcal{Y}$ , then $\mathcal{Y}$ is contained in a proper algebraic subgroup of ${\bf G}_m^n$ . Zilber's conjecture is stated more generally for semi-abelian varieties, and the conjecture above is open, although many partial results are known.

References

Primary source

Matt Bainbridge, Philipp Habegger and Martin Moeller, “Teichmueller curves in genus three and just likely intersections in G_m^n x G_a^n”, arXiv:1410.6835 (2014).

Posted 1 month ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Open

No proof or counterexample has been found; the conjecture remains open despite substantial partial results.

The conjecture, first posed by Boris Zilber and independently formulated by Bombieri, Masser, and Zannier in the early 2000s, asserts that sufficiently dense atypical intersections force a subvariety of an algebraic torus into a proper algebraic subgroup.

Known results

A weak form shows that atypical components are controlled by finitely many proper algebraic subgroups (Zilber, Kirby, Bombieri, Masser, and Zannier).
Partial cases are known for weak-transverse varieties in products of CM elliptic curves, including codimension two and certain relative-codimension-one cases.
Ax–Schanuel methods yield finite-control results, but not the full toric conjecture.

Current status (as of October 2026): The toric conjecture remains open; only weak forms and special cases are established, with no reported counterexample or complete proof.

Sources

No solutions have been posted yet.