20250909 Boaz Moerman

In arithmetic geometry, one of the most influential conjectures is Manin's conjecture. This conjecture gives an asymptotic for the number of rational points of bounded height on a Fano variety. More concretely, the conjecture gives a precise prediction for how many solutions a given polynomial equation has, for a broad class of such equations. Various instances of Manin's conjecture are known, but a proof is still out of reach.

In this talk, we introduce the framework of M-points on varieties, which encompasses many interesting subsets of rational points on these varieties. We extend Manin's conjecture to this more general setting, and prove this extension for (split) toric varieties. This theorem implies many concrete counting results, of which we will give several. For instance, it gives the likelyhood that the product of a triple of integers (x,y,z) is squareful.