NTS ABSTRACTFall2023

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Sept 7

Jiaqi Hou

Restrictions of eigenfunctions on arithmetic hyperbolic 3-manifolds

Let X be a compact arithmetic hyperbolic 3-manifold and Y a hyperbolic surface in X. Let f be a Hecke-Maass form on X, which is a joint eigenfunction of the Laplacian and Hecke operators. In this talk, I will present a power saving bound for the period of f along Y over the local bound. I will also present a work in progress on the bound for the L^2 norm of f restricted to Y. Both of the results are based on the method of arithmetic amplification developed by Iwaniec and Sarnak.

Sept 14

Ruofan Jiang

mod p analogue of Mumford-Tate and André-Oort conjectures for GSpin Shimura varieties

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, 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. Associated to it is the p-adic étale monodromy group. The Mumford-Tate conjecture claims that, the base change to Q_p of the Mumford-Tate group has the same neutral component with the p-adic étale monodromy group. On the other hand, 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.