In this paper, we give an explicit description of the first layer of anti-cyclotomic $\mathbf{Z}_{p}$-extension of imaginary quadratic fields.

We study the field isomorphism problem for a cubic generic polynomial $X^3+sX+s$ via Tschirnhausen transformation. Through this process, there naturally appears a $2$-dimensional involutive Cremona transformation. We show that the fixed field under the action of the transformation is purely transcendental over an arbitrary base field.

The indefinite orthogonal group $G = O(p,q)$ has a distinguished infinite dimensional unitary representation $\pi$, called the \textit{minimal representation} for $p+q$ even and greater than $6$. The \textit{Schr\"odinger model} realizes $\pi$ on a very simple Hilbert space, namely, $L^2(C)$ consisting of square integrable functions on a Lagrangean submanifold $C$ of the minimal nilpotent coadjoint orbit, whereas the $G$-action on $L^2(C)$ has not been well-understood. This paper gives an explicit formula of the unitary operator $\pi(w_0)$ on $L^2(C)$ for the `conformal inversion' $w_0$ as an integro-differential operator, whose kernel function is given by a Bessel distribution. Our main theorem generalizes the classic Schr\"odinger model on $L^2(\mathbf R^n)$ of the Weil representation, and leads us to an explicit formula of the action of the whole group $O(p,q)$ on $L^2(C)$. As its corollaries, we also find a representation theoretic proof of the inversion formula and the Plancherel formula for Meijer's $G$-transforms.

We give a uniqueness theorem of two meromorphic functions sharing two one-point sets and two two-point sets CM.

Let $\cal{F}$ be a family of meromorphic functions in a domain $D$, let $q, k$ be two positive integers, and let $a, b$ be two non-zero complex numbers. If, for each $f \in \cal {F}$, the zeros of $f $ have multiplicity at least $ k+1$, and $f=a \Leftrightarrow G(f)=b$, where $G(f)=P(f^{(k)})+H(f)$ be a differential polynomial of $f$ satisfying $q \geq \gamma_H$, and $\frac{\Gamma}{\gamma} |_H < k+1$, then $\cal {F}$ is normal in $D$.

We study the elliptic modular surface attached to the commutator subgroup of the modular group. This has an elliptic curve as base and only one singular fibre. We employ an algebraic approach and then consider some arithmetic questions.