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. This talk will also be available over zoom. ID: 93014934562 Password: The order of A9 (the alternating group of 9 elements)

Sept 21

Andreea Iorga

Realising certain semi-direct products as Galois groups

In this talk, I will prove that, under a specific assumption, any semi-direct product of a $p$-group $G$ with a group of order prime-to-$p$ $\Phi$ can appear as the Galois group of a tower of extensions $H/F/E$ with the property that $H$ is the maximal pro-$p$ extension of $F$ that is unramified everywhere, and $\Gal(H/F) = G$. At the end, I will show that a nice consequence of this is that any local ring admitting a surjection to $\mathbb{Z}_5$ or $\mathbb{Z}_7$ with finite kernel can be written as a universal everywhere unramified deformation ring.

Oct 05

Ziquan Yang

Arithmetic Deformation of Line Bundles

In the 70s, Deligne proved that any line bundle on a K3 surface in characteristic p > 0 lifts to characteristic 0 together with the surface. This theorem has played a fundamental role in the progress on the Tate conjecture for K3 surfaces in the past decades. In this talk, I will explain a generalization of Deligne's theorem, which states that in an arithmetric family, under some assumptions on the monodromy group and Kodaira-Spencer map, generically every line bundle in characteristic p deforms to characteristic 0. This is a joint work with David Urbanik.