On finiteness of odd superperfect numbers

Infinitary superperfect numbers

We shall show that $9$ is the only odd infinitary superperfect numbers.

A finiteness theorem for the Brauer group of K3 surfaces in odd characteristic

Let $k$ be a field finitely generated over the finite field $\mathbb F_p$ of odd characteristic $p$. For any K3 surface $X$ over $k$ we prove that the prime to $p$ component of the cokernel of the natural map $Br(k)\to Br(X)$ is finite.

On the finiteness of Carmichael numbers with Fermat factors and $L=2^αP^2$

Let $m$ be a Carmichael number and let $L$ be the least common multiple of $p-1$, where $p$ runs over the prime factors of $m$. We determine all the Carmichael numbers $m$ with a Fermat prime factor such that $L=2^αP^2$, where $k\in \mathbb{N}$ and $P$ is an odd prime number. There are eleven such Carmichael numbers.

整体函数域上模 $p$ Galois 表示的有限性定理

A finiteness theorem for mod $p$ Galois representations over global function fields

对于特征不同于 $p$ 的整体函数域 $K$，证明了 Artin 导子有界的连续几何半单表示 $\rho: G_K \to \mathrm{GL}_n(\overline{\mathbb{F}}_p)$ 只有有限多个同构类，且无需假设 $p$ 不整除 $n$。

Sporadic points of odd degree on $X_1(N)$ coming from $\mathbb{Q}$-curves

We say a closed point $x$ on a curve $C$ is sporadic if there are only finitely many points on $C$ of degree at most deg$(x)$. In the case where $C$ is the modular curve $X_1(N)$, most known examples of sporadic points come from elliptic curves with complex multiplication (CM). We seek to understand all sporadic points on $X_1(N)$ corresponding to $\mathbb{Q}$-curves, which are elliptic curves isogenous to their Galois conjugates. This class contains not only all CM elliptic curves, but also any elliptic curve $\overline{\mathbb{Q}}$-isogenous to one with a rational $j$-invariant, among others…

关于丢番图方程 $X^2+p^m=p_1^{n_1}p_2^{n_2}$

On the Diophantine equation $X^2+p^m=p_1^{n_1}p_2^{n_2}$

该研究针对含三个不同奇素数的丢番图方程 $X^2+p^m=p_1^{n_1}p_2^{n_2}$，通过二次域因式分解等方法得到相关指数的显式界，证明固定素数时方程解的个数有限。