2019
May 16 (Thur) 2019, 13:00~14:00, 아산이학관 526
Title: Congruence ideals and Tamagawa exponents
Speaker: Chan-Ho Kim (KIAS)
Abstract:
We discuss how much congruence ideals of modular forms vary under level lowering congruences. It turns out that the difference can be described in terms of local Tamagawa ideals as expected by Pollack and Weston. This is joint work with Kazuto Ota.
May 10 (Thur) 2019, 13:00~14:00, 아산이학관 526
Title: Spectrum of non-flat triangular drum, and the orbit of billiards on it: around the work of Akshey Venkatesh
Speaker: 임선희(서울대학교 수리과학부)
Abstract:
We will start with a general overview of interaction between number theory and dynamics. We will then look into the particular case of the subconvexity of L-function and explain its dynamical interpretation. We will then finish with the problems in Diophantine approximation and its dynamical solution. The last part is based on joint work with Uri Shapira and Nicolas de Saxcé.
May 2 (Thur) 2019, 15:00~16:00, 아산이학관 526
Title: Zero density estimates of an Epstein zeta function near the half line,
Speaker: Yoonbok Lee (Incheon National University)
Abstract:
Let $Q$ be a positive definite quadratic form with integral coefficients and let $E(s,Q)$ be the Epstein zeta function associated with $Q$. Assume that the class number of $Q$ is bigger than $1$. Then we estimate the number of zeros of $E(s,Q)$ in the region $ \Re s > \sigma_T ( \theta ) := 1/2 + ( \log T)^{- \theta}$ and $ T < \Im s < 2T$, to provide its asymptotic formula for fixed $ 0 < \theta < 1$ conditionally. Moreover, it is unconditional if the class number of $Q$ is $2$ or $3$ and $ 0 < \theta < 1/13$.
January 7 (Mon) 2019, 11:40~12:40, 아산이학관 526
Title: Effective equidistribution of rational points on certain expanding horospheres
Speaker: Min Lee (University of Bristol)
Abstract:
The main purpose of this talk is to provide an effective version of a result due to Einsiedler, Mozes, Shah and Shapira, on the equidistribution of rational points on expanding horospheres in the space of unimodular lattices in at least 3 dimensions. Their proof uses techniques from homogeneous dynamics and relies in particular on measure-classification theorems due to Ratner. Instead, we pursue an alternative strategy based on spectral theory, Fourier analysis and Weil’s bound for Kloosterman sums which yields an effective estimate on the rate of convergence in the space of (d + 1)-dimensional Euclidean lattices with d > 1. This extends my work with J. Marklof, on the 3-dimensional case (2017). This is a joint work with D. El-Baz and B. Huang.
January 7 (Mon) 2019, 10:30~11:30, 아산이학관 526
Title: Distribution of Hecke eigenvalues: large discrepancy
Speaker: Junehyuk Jung (Incheon National University)
Abstract:
Vertical Sato-Tate theorem for holomorphic modular forms concerns the distributionof eigenvalues of a fixed Hecke operator $T_p$ acting on the space of weight $k$ and level $N$ modular forms, as $k+N\to \infty$. It was proven by Serre (and independently by Sarnak) that there exists a limiting measure $\mu_p$, which depends only on $p$, such that the eigenvalues become equidistributed relatively to $\mu_p$.
Fix $N$ for simplicity. Then this can be restated in terms of the discrepancy between two measures: a probability measure $\mu_{p,k}$ supported on the eigenvalues of the
Hecke operator, and $\mu_p$, i.e., it is equivalent to $D(\mu_{p,k}, \mu_p) \to 0$. Regarding the rate of convergence, in the context of arithmetic quantum chaos, it was suggested both by speculation and numerical test that
\[
D(\mu_{p,k}, \mu_p) =O(k^{-1/2+\epsilon}).
\]
In this talk, I'm going to disprove this by showing that
\[
D(\mu_{p,k}, \mu_p) =\Omega(k^{-1/3}\log^2 k).
\]
This is a joint work with Naser Talebizadeh Sardari and Simon Marshall.