If X/S is a family of smooth projective varieties over \overline{Q}, then the Hodge filtration on the cohomology of X induces locally an analytic period map from S(C) to a flag variety. The flag variety also has a natural algebraic structure over \overline{Q}, and it is natural to ask: which (\overline{Q}-)algebraic conditions on the Hodge filtration induce (\overline{Q}-)algebraic conditions on S?

If the Hodge conjecture holds, then the condition that a given rational cohomology class on a tensor power of the cohomology be a Hodge cycle, which is evidently algebraic on the flag variety, is also \overline{Q}-algebraic on S (if we only ask that it be C-algebraic then this is a theorem of Cattani-Deligne-Kaplan). Moreover, if the Grothendieck Period Conjecture also holds, then using a result of Andre on the existence of an \overline{Q}-rational Hodge generic point it can be shown that these are essentially the only ones. For families of abelian varieties, there is an unconditional proof of this bialgebraicity theorem due to Ullmo-Yafaev.

In joint work in progress with Christian Klevdal, we investigate a local p-adic analytic analog of this story: now X/S is a smooth proper family of rigid analytic varieties defined over a p-adic field, and we ask when rigid analytic conditions on the Hodge-Tate filtration on p-adic etale cohomology induce rigid analytic conditions on S. In this talk I will explain some of our results, with an emphasis on the connection between our strategy of proof and the conjectural strategy described above over Q.