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.

Oct 12

Baiqing Zhu

Arithmetic Siegel-Weil formula on the modular curve X_0(N)

The arithmetic Siegel-Weil formula relates the Siegel Eisenstein series and intersection numbers of special cycles on Shimura varieties. In this talk, we focus on the modular curve X_0(N) and give a proof of the arithmetic Siegel-Weil formula on X_0(N) for both of the nonsingular and singular Fourier coefficients of an explicit Eisenstein series. Our proof is based on the difference formulas of both the geometric side and the analytic side, and also the previous works of Yang, Shi, Sankaran, and Du.

Oct 26

Simon Marshall

Large values of eigenfunctions on hyperbolic manifolds

It is a folklore conjecture that the sup norm of a Laplace eigenfunction on a compact hyperbolic surface grows more slowly than any positive power of the eigenvalue. In dimensions three and higher, this was shown to be false by Iwaniec-Sarnak and Donnelly, using the theta lift. I will present joint work with Farrell Brumley that strengthens these results, and extends them to locally symmetric spaces associated to SO(p,q).

Nov 2

Tonghai Yang

Regularized theta lifting and Complex Multiplication

In the 1980s, Gross and Zagier proved the famous Gross-Zagier formula, their beautiful factorization formula of singular moduli, and gave a deep conjecture on algebraicity of CM values of higher Green functions. In this talk, we use regularized theta lifting to give another look at their work and extend them to Shimura varieties. We also give a proof of their algebraicity conjecture. Part of this talk is joint work with Jan Bruinier and Yingkun Li.

Nov 9

Tung Nguyen

On the arithmetic of Fekete polynomials of principal Dirichlet characters

Fekete polynomials have a rich history in mathematics. They first appeared in the work of Michael Fekete in his investigation of Siegel zeros of Dirichlet L-functions. In a previous study, we explored the arithmetic of generalized Fekete polynomials associated with primitive quadratic Dirichlet characters. We found that these polynomials possess a variety of interesting and important arithmetic and Galois-theoretic properties. In this talk, we will introduce a different incarnation of Fekete polynomials, namely those associated with principal Dirichlet characters. Through numerical experiments, we examine their cyclotomic and non-cyclotomic factors and identify some of their roots in the unit circle. We also investigate their modular properties and special values. Last but not least, based on both theoretical and numerical data, we propose a precise question on the structure of the Galois group of these Fekete polynomials. This is based on joint work with Shiva Chidambaram, Jan Minac, and Nguyen Duy Tan.

Nov 16

Maksym Radziwill

Gap disribution of \sqrt{n} modulo 1 and the circle method

Let \alpha be a positive non-integer real number. We consider the values n^{\alpha} modulo 1 with n < N and place them in order on the interval [0,1]. It is tempting to conjecture that these values should look like a set of random points, therefore the gap distribution (at the scale 1/N) should be Poisson. It turns out however, as proven by Elkies and McMullen, that the gap distribution exists and is not Poisson when \alpha = 1/2. This is the only exponent for which the existence of the gap distribution is proven. The proof of Elkies-McMullen and the later proof of Browning-Vinogradov are dynamical in nature. We will present a self-contained and elementary proof based on the circle method. I will discuss the differences between the two proofs, time permitting. Joint work with Niclas Technau.

Dec 7

Chengyang Bao

Computing crystalline deformation rings via the Taylor--Wiles--Kisin patching method

Consider an odd prime $p$ and a fixed absolutely irreducible representation $\rbar: G_{\Q_p}\to \GL_2(\F_p)$. Let $R^k$ denote the non-framed fixed-determinant crystalline deformation ring of $\rbar$, whose $\overline{\Q}_p$-points parametrize the crystalline representations of Hodge--Tate weights $(0,k-1)$ that reduce to $\rbar$ modulo $p$. In this talk, we will outline an algorithm designed to compute arbitrarily close approximations of $R^k$ and, consequently, approximations of the Hilbert series of $R^k/p$. We will begin by explaining the significance of these rings, recalling their role in modularity lifting theorems. Following this, we will provide a short survey of the current understanding of crystalline deformation rings. Finally, we will discuss the algorithm's design, with a discussion on the collected data if time permits.