On Special Subvarieties of the Universal Semi-abelian Scheme and Zilber--Pink Conjectures

Abstract

We classify the special subvarieties of the universal semi-abelian scheme of arbitrary toric rank, viewed as a mixed Shimura variety. As an application, we show that the Zilber--Pink conjecture for mixed Shimura varieties implies its counterpart for semi-abelian varieties, correcting an error in an unpublished manuscript of Pink. Our classification also reveals that the relative Manin--Mumford conjecture for semi-abelian schemes must account for subvarieties of Ribet type, and we give a corrected formulation that does so.

61 pages. Substantially rewritten and expanded. Theorem 1.4 in previous versions is not correct as stated. We now work directly in arbitrary toric rank