이원웅 (카이스트)

Algebraic independence of the Eisenstein series for the arithmetic Hecke groups

초록:

Let $E_{2k}$ be the Eisenstein series of weight $2k$ for $\mathrm{SL}_2(\mathbb Z)$. Nesterenko shows that for any $\alpha \in \mathbb H$, at least three of the numbers $e^{2\pi i \alpha},E_2(\alpha),E_4(\alpha),E_6(\alpha)$ are algebraically independent. We prove the analogous result for the arithmetic Hecke groups. Namely, we show that for $m=3,4,6$, at least three of the numbers $e^{2\pi i \alpha},E_{2,m}(\alpha),E_{4,m}(\alpha),E_{6,m}(\alpha)$ are algebraically independent, where $E_{2k,m}$ is the Eisenstein series of weight $2k$ for the arithmetic Hecke group $H(m)$.

권재성 (유니스트)

Modular symbols on Bianchi manifolds

초록: Bianchi manifold is a 3-dimensional real hyperbolic manifold related to the automorphic forms of GL(2) over imaginary quadratic fields. We study the distribution of modular symbols on Bianchi manifolds. The main idea is estimating error terms of the value of the integral related to the special modular symbols.

이영민 (카이스트)

Classification of Modular form of half-integral weight in $S^{+}_{\lambda+\frac{1}{2}}(\Gamma_0(4))$ with few non-vanishing coefficients modulo $\ell$

초록:

Let $f(z)=\sum_{n=1}^{\infty} a(n)q^n\in S^{+}_{\lambda+\frac{1}{2}}(\Gamma_0(4))\cap \mathcal{O}_{K}[[q]]$ with some number field $K$. Let $\ell$ be a prime larger than $3$ and $v$ be a prime of $K$ over $\ell$. Assume that

\[f(z)\equiv \sum_{i=1}^{t} \sum_{m=1}^{\infty} a(n_im^2)q^{n_im^2} \pmod{v} \]

with some square-free integers $n_i$. Then, $f(z)$ can be expressed as linear combination of Hecke eigenforms with few non-vanishing coefficients modulo $\ell$.