On the powerful values of polynomials over number fields

In this paper, assuming a conjecture of Vojta on the bounded degree algebraic numbers on a number field $k$, we determine explicit lower and upper bounds for the cardinal number of the set of polynomials $f\in k[x]$ with degree $r\geq 2$ whose irreducible factors have multiplicity strictly less than $s$ and the values $f(b_1),\cdots, f(b_M)$ are $s$-powerful elements in $k^*$ for a certain positive integer $M$, where $b_i$'s belong to an arbitrary sequence of the pairwise distinct element of $k$ that satisfy certain conditions

Twists of the Albanese varieties of cyclic multiple planes with large ranks over higher dimension function fields

In [17], we proved a structure theorem on the Mordell-Weil group of abelian varieties over function fields that arise as the twists of abelian varieties by the cyclic covers of projective varieties in terms of the Prym varieties associated with covers. In this paper, we provide an explicit way to construct the abelian varieties with large ranks over the higher dimension function fields. To do so, we apply the above-mentioned theorem to the twists of Albanese varieties of the cyclic multiple planes

Vojta's conjecture on weighted projective varieties and an application on GCD's

We state Vojta's conjecture for smooth weighted projective varieties, weighted multiplier ideal sheaves, and weighted log pairs and prove that all three versions of the conjecture are equivalent. Moreover, we introduce generalized weighted general common divisors and express them as heights of weighted projective spaces blown-up at a point, relative to an exceptional divisor. Furthermore, we also prove that assuming Vojta's conjecture for weighted projective varieties one can bound the $\log {h_{wgcd} \,}$ for any subvariety of codimension $\geq 2$ and a finite set of places $S$. An analogue result is proved for weighted homogeneous polynomials with integer coefficients. As an application of our results we obtain a bound on greatest common divisors, which restricted to projective space is the same as bounds obtained by Corvaja, Zannier, et al

Local and global heights on weighted projective varieties and Vojta's conjecture

In this paper we extend results on weighted heights by extending the definition of weighted heights and introducing the Weil machinery for weighted projective spaces via Cartier and Weil divisors. We define such heights for weighted projective varieties and their closed subvarieties. Moreover, we state Vojta's conjecture for weighted projective varieties in terms of weighted heights and generalized weighted greatest common divisors.