Let $n$ be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if $n>1$ and both $6n^2-1$ and $12n^2+1$ are odd primes, then the general elliptic curve $y^2=x^3+(36n^2-9)x-2(36n^2-5)$ has only the integral point $(x,y)=(2,0)$. By this result we can get that the above elliptic curve has only the trivial integral point for $n=3,13,17$ etc. Thus it can be seen that the elliptic curve $y^2=x^3+27x-62$ really is an unusual elliptic curve which has large integral points.

Let $k$ be an algebraically closed field of characteristic $p\>0$. We study obstructions to lifting to characteristic $0$ the faithful continuous action $\phi$ of a finite group $G$ on $k\left[\right[t\left]\right]$. To each such $\phi$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $\phi$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic $0$ in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H\subset G$. We determine for which $G$ the KGB obstruction...

Let $X$ be a curve over a field $k$ with a rational point $e$. We define a canonical cycle ${\Delta }_e\in Z^2(X^3{)}_{\mathrm{hom}}$. Suppose that $k$ is a number field and that $X$ has semi-stable reduction over the integers of $k$ with fiber components non-singular. We construct a regular model of $X^3$ and show that the height pairing $⟨{\tau }_{*}\left({\Delta }_e\right),{\tau }_{*}^{\text{'}}\left({\Delta }_e\right)⟩$ is well defined where $\tau$ and ${\tau }^{\text{'}}$ are correspondences. The paper ends with a brief discussion of heights and $L$-functions in the case that $X$ is a modular curve.

In this paper we show that for every prime $p\ge 5$ the dimension of the $p$-torsion in the Tate-Shafarevich group of $E/K$ can be arbitrarily large, where $E$ is an elliptic curve defined over a number field $K$, with $[K:\mathbb{Q}]$ bounded by a constant depending only on $p$. From this we deduce that the dimension of the $p$-torsion in the Tate-Shafarevich group of $A/\mathbb{Q}$ can be arbitrarily large, where $A$ is an abelian variety, with $dimA$ bounded by a constant depending only on $p$.