Unlikely intersections for curves in products of Carlitz modules

Abstract

The conjectures associated with the names of Zilber-Pink greatly generalize results associated with the names of Manin-Mumford and Mordell-Lang, but unlike the latter they are almost exclusively restricted to zero characteristic. Not so long ago the second author made a start on removing this restriction by studying multiplicative groups over positive characteristic, and recently both authors went further for additive groups with extra Frobenius structure. Here we study additive groups with extra structure coming instead from the Carlitz module. We state a conjecture for curves in general dimension and we prove it in three dimensions. The main tool is a new relative version (for cyclotomic fields) of Denis’s analogue of Dobrowolski’s classical lower bound for heights, as well as a suitable upper bound. We also work out a couple of special cases in two dimensions: for example with respect to prime fields there are exactly 23 Carlitz roots of unity whose reciprocals are also roots of unity.