arXiv subjects

Elyes Boughattas

Publications and source records attributed to Elyes Boughattas.

3 recordsLinked to original sources

arXiv2024

Unirationality and $R$-equivalence for conic bundles over quasi-finite fields

Yanchevskiĭ had asked whether conic bundle surfaces over $\mathbf{P}^1_k$ are unirational when $k$ is a finite field. We give a partial answer to his question by showing that for quasi-finite fields $k$ (e.g. finite fields) a regular conic bundle $X$ over $\mathbf{P}^1_k$ is unirational if all non-split fibres lie over rational points. For large finite fields $k$, this beats a previous result of Mestre. Under the same assumption, we also prove that all rational points of $X$ are $R$-equivalent.

math.AG↗

arXiv2024

Pureté de l'approximation forte sur le corps des fonctions d'une courbe algébrique complexe

Over the function field of a complex algebraic curve, strong approximation off a non-empty finite set of places holds for the complement of a codimension $2$ closed subset in a homogeneous space under a semisimple algebraic group, and for the complement of a codimension $2$ closed subset in an affine smooth complete intersection of low degree.

math.AG↗

arXiv2024

The Grunwald problem and homogeneous spaces with non-solvable stabilisers

We give an affirmative answer to the Grunwald problem for new families of non-solvable finite groups G, away from the set of primes dividing |G|. Furthermore, we show that such G verify the condition (BM), that is, the Brauer-Manin obstruction to weak approximation is the only one for quotients of SL_n by G. These new families include extensions of groups satisfying (BM) by kernels which are products of symmetric groups S_m, with $m\neq 2,6$, and alternating groups A_5. We also investigate (BM) for small groups by giving an explicit list of small order groups for which (BM) is unknown and we show that for many of them (BM) holds under Schinzel's hypothesis.

math.NT↗