arXiv subjects

Abdulmuhsin Alfaraj

Publications and source records attributed to Abdulmuhsin Alfaraj.

4 recordsLinked to original sources

arXiv2025

Brauer groups of certain affine cubic surfaces

We study the Brauer groups of affine surfaces that are complements of singular hyperplane sections of smooth cubic surfaces over a field $k$ of characteristic $0$. We determine the Brauer group over the algebraic closure as a Galois module for all the possible singular hyperplane sections. For the case when the hyperplane section is geometrically the union of three lines, we give explicit examples where transcendental elements of order $2$ and $3$ exist over $\mathbb{Q}$. We end with an application on the integral Brauer-Manin obstruction to the integral Hasse principle.

math.AG↗

arXiv2025

Brauer groups of conic bundles over elliptic curves

We study the Brauer groups of regular conic bundles over elliptic curves defined over a number field $k$. We explicitly compute the Brauer group of the conic bundle when the singular fibres lie above $k$-points that are divisible by $2$ in $E(k)$, and the corresponding ramification fields are isomorphic. We apply the result to compute the Brauer group of a class of surfaces analogous to that of Châtelet surfaces. We investigate Brauer-Manin obstructions to weak approximation coming from the real places on such surfaces.

math.AG↗

arXiv2025

Manin's Conjecture for Equivariant compactifications of forms of $\mathbb{G}_a^n$

We prove the Batyrev-Manin conjecture for smooth equivariant compactifications of forms of $\mathbb{G}_a^n$ over a global function field $F$, assuming some conditions on the boundary divisor. To verify that the leading constant agrees with Peyre's predicition we also show that a commutative unipotent group admitting a smooth equivariant compactification satisfies the Hasse principle for algebraic groups and weak approximation. We study in detail the case of $\mathbb{P}^{p-1}$, where $p$ is the characteristic of $F$, viewed as a compactification of appropriate $F$-wound groups to illustrate new phenomena appearing in the function field setting.

math.NT↗

arXiv2021

On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences

We prove that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on elliptic curves of the from $y^2=x(x^2+b)$, where $b$ is any positive integer. We achieve this by using the modularity of elliptic curves over real quadratic number fields.

math.NT↗