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Evis Ieronymou

Publications and source records attributed to Evis Ieronymou.

8 recordsLinked to original sources
arXiv2023

Odd torsion Brauer elements and arithmetic of diagonal quartic surfaces over number fields

We use recent advances in the local evaluation of Brauer elements to study the role played by {\it odd} torsion elements of the Brauer group in the arithmetic of diagonal quartic surfaces over {\it arbitrary} number fields. We show that over a local field if the order of the Brauer element is odd and coprime to the residue characteristic then the evaluation map it induces on the local points is constant. Over number fields we give a sufficient condition on the coefficients of the equation, which is mild and easy to check, so that the odd torsion does not obstruct weak approximation. We also note a systematic way to produce $K3$ surfaces over $\Q_2$ with good reduction and a non-trivial 2-torsion element of the Brauer group with Swan conductor zero.

math.NT↗
arXiv2022

Brauer-Manin obstruction for zero-cycles on certain varieties

We investigate the question of whether the existence of a family of local zero-cycles of degree $d$ orthogonal to the Brauer group implies the non-emptiness of the Brauer-Manin set for certain varieties. We provide various examples of Brauer-Manin obstruction to the existence of zero-cycles of appropriate degrees.

math.AG↗
arXiv2022

Applications of the fibration method for zero-cycles to the Brauer-Manin obstruction to the existence of zero-cycles on certain varieties

We study the Brauer-Manin obstruction to the existence of zero-cycles of degree $d$ on certain classes of varieties over number fields. We generalise existing results in the literature and prove some results about fibrations over the projective line, where the geometric Brauer group of the generic fibre is not assumed to be finite. The idea is to assume that the Brauer-Manin obstruction to the Hasse principle is the only one for certain fibres and then deduce analogous results for zero-cycles.

math.AG↗
arXiv2011

On the Brauer group of diagonal quartic surfaces

We obtain an easy sufficient condition for the Brauer group of a diagonal quartic surface D over Q to be algebraic. We also give an upper bound for the order of the quotient of the Brauer group of D by the image of the Brauer group of Q. The proof is based on the isomorphism of the Fermat quartic surface with a Kummer surface due to Masumi Mizukami.

math.AG↗
arXiv2009

Diagonal quartic surfaces and transcendental elements of the Brauer group

We exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic surface over a number field is algebraic and give sufficient conditions for this to be the case. In the last section we give an obstruction to weak approximation due to a transendental class on a specific diagonal quartic surface, an obstruction which cannot be explained by the algebraic Brauer group which in this case is just the constant algebras.

math.NT↗