Tate-linear subvarieties and p-adic monodromy We work over an algebraically closed field of characteristic p>0. A Tate-linear subvariety in the moduli space of ordinary principally polarized abelian varieties is one whose formal completion at one (or every) point is a formal subtorus of the Serre-Tate deformation space. These special subvarieties arise in connection with the Hecke orbit problem. We introduce a global version of the Serre-Tate coordinates to study them, and their p-adic monodromy. A major question is whether every Tate-linear subvariety is the reduction of a Shimura subvariety. We formulate several conjectures, the strongest one is an analogue of the Mumford-Tate conjecture. If time permits we will also discuss some evidence of the conjectures, together with an application to p-adic L-functions of the techniques introduced here.