Number Theory Seminar

Mumford-Tate and André-Oort conjectures are two influential problems which have been studied for decades. The conjectures are originally stated in char 0. For a given smooth projective variety Y over complex numbers, one has the notion of Hodge structure. Associated to the Hodge structure is a Q reductive group MT(Y), called the Mumford-Tate group. If the variety is furthermore defined over a number field, then its p-adic étale cohomology is a Galois representation. Then there is a notion of p-adic étale monodromy group G_p(Y). The Mumford-Tate conjecture claims that the base change to Q_p of MT(Y) has the same neutral component with G_p(Y). The André-Oort conjecture claims that, if a subvariety of a Shimura variety contains a Zariski dense collection of special points, then the subvariety is itself a Shimura subvariety.

My talk will be on my recent work on mod p analogues of the conjectures for mod p GSpin Shimura varieties. Important special cases of GSpin Shimura varieties include moduli spaces of polarized Abelian and K3 surfaces.