NTS ABSTRACTSpring2019

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Jan 23

Yunqing Tang

Feb 1

Yunqing Tang

The diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$

Abstract: Assume a polynomial-time algorithm for factoring integers, Conjecture~\ref{conj}, $d\geq 3,$ and $q$ and $p$ prime numbers, where $p\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\log(q)$ that lifts every $\mathbb{Z}/q\mathbb{Z}$ point of $S^{d-2}\subset S^{d}$ to a $\mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 \text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$.

Feb 8

Roman Fedorov

A conjecture of Grothendieck and Serre on principal bundles in mixed characteristic

Abstract: Let G be a reductive group scheme over a regular local ring R. An old conjecture of Grothendieck and Serre predicts that such a principal bundle is trivial, if it is trivial over the fraction field of R. The conjecture has recently been proved in the "geometric" case, that is, when R contains a field. In the remaining case, the difficulty comes from the fact, that the situation is more rigid, so that a certain general position argument does not go through. I will discuss this difficulty and a way to circumvent it to obtain some partial results.

Feb 13

Frank Calegari

Recent Progress in Modularity

Abstract: We survey some recent work in modularity lifting, and also describe some applications of these results. This will be based partly on joint work with Allen, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne, and also on joint work with Boxer, Gee, and Pilloni.

Feb 15

Junho Peter Whang

Integral points and curves on moduli of local systems

Abstract: We consider the Diophantine geometry of moduli spaces for special linear rank two local systems on surfaces with fixed boundary traces. After motivating their Diophantine study, we establish a structure theorem for their integral points via mapping class group descent, generalizing classical work of Markoff (1880). We also obtain Diophantine results for algebraic curves in these moduli spaces, including effective finiteness of imaginary quadratic integral points for non-special curves.

Feb 22

Yifan Yang

Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus

Abstract: In this talk we consider the rational torsion subgroup of the generalized Jacobian of the modular curve X_0(N) with respect to a reduced divisor given by the sum of all cusps. When N=p is a prime, we find that the rational torsion subgroup is always cyclic of order 2 (while that of the usual Jacobian of X_0(p) grows linearly as p tends to infinity, according to a well-known result of Mazur). Subject to some unproven conjecture about the rational torsions of the Jacobian of X_0(p^n), we also determine the structure of the rational torsion subgroup of the generalized Jacobian of X_0(p^n). This is a joint work with Takao Yamazaki.

March 22

Fang-Ting Tu

Title: Supercongrence for Rigid Hypergeometric Calabi-Yau Threefolds

Abstract: This is a joint work with Ling Long, Noriko Yui, and Wadim Zudilin. We establish the supercongruences for the rigid hypergeometric Calabi-Yau threefolds over rational numbers. These supercongruences were conjectured by Rodriguez-Villeagas in 2003. In this work, we use two different approaches. The first method is based on Dwork's p-adic unit root theory, and the other is based on the theory of hypergeometric motives and hypergeometric functions over finite fields. In this talk, I will introduce the first method, which allows us to obtain the supercongruences for ordinary primes.

April 12

Junehyuk Jung

Title: Quantum Unique Ergodicity and the number of nodal domains of automorphic forms

Abstract: It has been known for decades that on a flat torus or on a sphere, there exist sequences of eigenfunctions having a bounded number of nodal domains. In contrast, for a manifold with chaotic geodesic flow, the number of nodal domains of eigenfunctions is expected to grow with the eigenvalue. In this talk, I will explain how one can prove that this is indeed true for the surfaces where the Laplacian is quantum uniquely ergodic, under certain symmetry assumptions. As an application, we prove that the number of nodal domains of Maass-Hecke eigenforms on a compact arithmetic triangles tends to $+\infty$ as the eigenvalue grows. I am going to also discuss the nodal domains of automorphic forms on $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$. Under a minor assumption, I will give a quick proof that the real part of weight $k\neq 0$ automorphic form has only two nodal domains. This result captures the fact that a 3-manifold with Sasaki metric never admits a chaotic geodesic flow. This talk is based on joint works with S. Zelditch and S. Jang.

April 19

Hang Xue (Arizona)

Title: Arithmetic theta lifts and the arithmetic Gan--Gross--Prasad conjecture.

Abstract: I will explain the arithmetic analogue of the Gan--Gross--Prasad conjecture for unitary groups. I will also explain how to use arithmetic theta lift to prove certain endoscopic cases of it.

May 3

Matilde Lalin (Université de Montréal)

Title: The mean value of cubic $L$-functions over function fields.

Abstract: We will start by exploring the problem of finding moments for Dirichlet $L$-functions, including the first main results and the standard conjectures. We will then discuss the problem for function fields. We will then present a result about the first moment of $L$-functions associated to cubic characters over $\F_q(t)$, when $q\equiv 1 \bmod{3}$. The case of number fields was considered in previous work, but never for the full family of cubic twists over a field containing the third roots of unity. This is joint work with C. David and A. Florea.

May 10

Hector Pasten (Harvard University)

Title: Shimura curves and estimates for abc triples.

Abstract: I will explain a new connection between modular forms and the abc conjecture. In this approach, one considers maps to a given elliptic curve coming from various Shimura curves, which gives a way to obtain unconditional results towards the abc conjecture starting from good estimates for the variation of the degree of these maps. The approach to control this variation of degrees involves a number of tools, such as Arakelov geometry, automorphic forms, and analytic number theory. The final result is an unconditional estimate that lies beyond the existing techniques in the context of the abc conjecture, such as linear forms in logarithms.