Publications
88 results found
Lazda CD, Skorobogatov AN, 2023, Reduction of Kummer surfaces modulo 2 in the non-supersingular case, EPIJOURNAL DE GEOMETRIE ALGEBRIQUE, Vol: 7, ISSN: 2491-6765
Skorobogatov AN, Sofos E, 2023, Schinzel Hypothesis on average and rational points, Inventiones Mathematicae, Vol: 231, Pages: 673-739, ISSN: 0020-9910
We resolve Schinzel’s Hypothesis (H) for 100% of polynomials of arbitrary degrees. We deduce that a positive proportion of diagonal conic bundles over Q with any given number of degenerate fibres have a rational point, and obtain similar results for generalised Châtelet equations.
Skorobogatov AN, Valloni D, 2022, ENRIQUES INVOLUTIONS AND BRAUER CLASSES, NAGOYA MATHEMATICAL JOURNAL, ISSN: 0027-7630
Orr M, Skorobogatov A, Valloni D, et al., 2022, Invariant Brauer group of an abelian variety, Israel Journal of Mathematics, Vol: 249, Pages: 695-733, ISSN: 0021-2172
We study a new object that can be attached to an abelian variety or a complex torus: the invariant Brauer group, as recently defined by Yang Cao. Over the field of complex numbers this is an elementary abelian 2-group with an explicit upper bound on the rank. We exhibit many cases in which the invariant Brauer group is zero, and construct complex abelian varieties in every dimension starting with 2, both simple and non-simple, with invariant Brauer group of order 2. We also address the situation in finite characteristic and over non-closed fields.
Gvirtz D, Skorobogatov A, 2022, Cohomology and the Brauer groups of diagonal surfaces, Duke Mathematical Journal, Vol: 171, Pages: 1299-1347, ISSN: 0012-7094
We present a method for calculating the Brauer group of a surface given by a diagonal equation in projective space. For diagonal quartic surfaces with coefficients in Q we determine the Brauer groups over Q and Q(i)
Colliot-Thélène JL, Skorobogatov AN, 2021, Varieties with a group action, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 215-229
One often needs to study the Brauer group of a variety equipped with an action of an algebraic group. The Brauer groups of connected algebraic groups themselves as well as the Brauer groups of their homogeneous spaces can be explicitly computed in many cases.
Colliot-Thélène JL, Skorobogatov AN, 2021, Brauer groups of schemes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 71-99
There are two ways to generalise the Brauer group of fields to schemes. The definition of the Brauer group of a field k in terms of central simple algebras over k readily extends to schemes as the group of equivalence classes of Azumaya algebras.
Colliot-Thélène JL, Skorobogatov AN, 2021, Rationality in a family, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 295-303
The specialisation method allows one to prove that a smooth and projective complex variety is not stably rational if it can be deformed into a mildly singular variety Z whose desingularisation has a non-zero Brauer group.
Colliot-Thélène JL, Skorobogatov AN, 2021, The Tate conjecture, abelian varieties and K3 surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 395-425
M. Artin and J. Tate conjectured that the Brauer group of a smooth and projective variety over a finite field is a finite group. In his 1966 Bourbaki talk [Tate66b], Tate explains why this is analogous to the conjectured finiteness of the Tate–Shafarevich group of an abelian variety over a number field.
Colliot-Thélène JL, Skorobogatov AN, 2021, Comparing the two Brauer groups, II, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 101-120
Our goal here is to give a very short list of key concepts with some examples. This is not a replacement for a detailed introduction to stacks, algebraic spaces and gerbes, for which we refer the reader to a very helpful book by Olsson [Ols16], see also [SGA1, Ch. VI], [Gir71], [LMB00], [Vis05] and [Stacks].
Colliot-Thélène JL, Skorobogatov AN, 2021, Varieties over a field, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 121-163
In this chapter we describe a general technique for computing the Brauer group Br(X) of a smooth projective variety X over a field k.
Colliot-Thélène JL, Skorobogatov AN, 2021, Schemes over local rings and fields, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 231-262
The object of study in this chapter is a scheme over the spectrum of a local ring. A separately standing Section 10.1 is devoted to the concepts of a split variety and of a split fibre of a morphism of varieties; for arithmetic applications and for the calculation of the Brauer group, split fibres should be considered as ‘good’ or ‘non-degenerate’.
Colliot-Thélène JL, Skorobogatov AN, 2021, Étale cohomology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 43-70
The étale topology and étale cohomology were invented by A. Grothendieck in the beginning of the 1960s, after Serre’s discussion of local triviality for principal homogeneous spaces in algebraic geometry [Ser58].
Colliot-Thélène JL, Skorobogatov AN, 2021, The Brauer–Manin obstruction for zero-cycles, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 381-393
The Brauer–Manin obstruction for rational points has an analogue for zerocycles, which conjecturally governs the local-to-global principle for zero-cycles on an arbitrary smooth projective variety X – unlike the original version for rational points! For example, one expects that if X has a family of local zerocycles of degree 1 for each completion of k, which is orthogonal to Br(X) with respect to the Brauer–Manin pairing, then X has a global zero-cycle of degree 1.
Colliot-Thélène JL, Skorobogatov AN, 2021, Galois cohomology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 3-42
This chapter begins with a brief introduction to quaternion algebras over a field. After recalling basic facts about central simple algebras, we discuss the classical definition of the Brauer group of a field as the group of equivalence classes of such algebras.
Colliot-Thélène JL, Skorobogatov AN, 2021, Are rational points dense in the Brauer–Manin set?, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 345-380
Let X be a smooth, projective and geometrically integral variety over a number field k.
Colliot-Thélène JL, Skorobogatov AN, 2021, Birational invariance, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 165-179
In Section 6.2 we discuss the unramified Brauer group Brnr(K/k) ⊂ Br(K) of a field K over a subfield k.
Colliot-Thélène JL, Skorobogatov AN, 2021, Severi–Brauer varieties and hypersurfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 181-197
There is a natural bijection between the isomorphism classes of Severi–Brauer varieties over a field k and the isomorphism classes of central simple k-algebras.
Colliot-Thélène JL, Skorobogatov AN, 2021, The Brauer group and families of varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 263-293
In this section we are interested in the following question. Let f : X→Y be a dominant morphism of regular integral varieties over a field k.
Colliot-Thélène JL, Skorobogatov AN, 2021, The Brauer–Manin set and the formal lemma, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 305-344
This is the first of three chapters which deal with applications of the Brauer group to the arithmetic of varieties over a number field k.
Colliot-Thélène JL, Skorobogatov AN, 2021, Singular schemes and varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, Pages: 199-214
This chapter collects and in some cases rectifies a number of results in the literature on the Brauer groups of singular schemes.
Colliot-Thélène J-L, Skorobogatov AN, 2021, The Brauer–Grothendieck Group, Publisher: Springer International Publishing, ISBN: 9783030742478
Loughran D, Skorobogatov A, Smeets A, 2020, Pseudo-split fibres and arithmetic surjectivity, Annales Scientifiques de l'École Normale Supérieure, Vol: 53, Pages: 1037-1070, ISSN: 0012-9593
Let f : X → Y be a dominant morphism of smooth, proper and geometrically integral varieties over a number field k, with geometricallyintegral generic fibre. We give a necessary and sufficient geometric criterion for the induced map X ( k v ) → Y ( k v ) to be surjective for almost all places v of k. This generalises a result of Denef which had previously been conjectured by Colliot-Th ́el`ene, and can be seen as an optimal geometric version of the celebrated Ax–Kochen theorem.
Orr M, Skorobogatov A, Zarhin Y, 2019, On uniformity conjectures for abelian varieties and K3 surfaces, American Journal of Mathematics, Vol: 143, ISSN: 0002-9327
We discuss logical links among uniformity conjectures concerning K3 sur-faces and abelian varieties of bounded dimension defined over number fields of bounded degree. The conjectures concern the endomorphism algebra of an abelian variety, the Neron–Severi lattice of a K3 surface, and the Galois invariant subgroup of the geometric Brauer group.
Creutz B, Viray B, Skorobogatov AN, 2019, Degree and the Brauer-Manin obstruction, ALGEBRA & NUMBER THEORY, Vol: 12, Pages: 2445-2470, ISSN: 1937-0652
Skorobogatov AN, Orr M, 2018, Finiteness theorems for K3 surfaces and abelian varieties of CM type, Compositio Mathematica, Vol: 154, Pages: 1571-1592, ISSN: 0010-437X
We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford–Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of Várilly-Alvarado in the CM case.
Skorobogatov AN, Zarhin Y, 2018, Kummer varieties and their Brauer groups, Pure and Applied Mathematics Quarterly, Vol: 13, Pages: 337-368, ISSN: 1558-8599
We study Kummer varieties attached to 2-coverings of abelian varieties ofarbitrary dimension. Over a number field we show that the subgroup of oddorder elements of the Brauer group does not obstruct the Hasse principle.Sufficient conditions for the triviality of the Brauer group are given, whichallow us to give an example of a Kummer K3 surface of geometric Picardrank 17 over the rationals with trivial Brauer group. We establish the non-emptyness of the Brauer–Manin set of everywhere locally soluble Kummervarieties attached to 2-coverings of products of hyperelliptic Jacobians withlarge Galois action on 2-torsion.
Skorobogatov AN, 2017, Cohomology and the Brauer group of double covers, Brauer Groups and Obstruction Problems Moduli Spaces and Arithmetic, Publisher: Birkhäuser, ISBN: 9783319468525
Ieronymou E, Skorobogatov AN, 2017, Odd order Brauer-Manin obstruction on diagonal quartic surfaces (vol 270, pg 181, 2015), ADVANCES IN MATHEMATICS, Vol: 307, Pages: 1372-1377, ISSN: 0001-8708
Harpaz Y, Skorobogatov AN, 2016, Hasse principle for Kummer varieties, Algebra and Number Theory, Vol: 10, Pages: 813-841, ISSN: 1944-7833
The existence of rational points on the Kummer variety associated to a 22-covering of an abelian variety AA over a number field can sometimes be established through the variation of the 22-Selmer group of quadratic twists of AA. In the case when the Galois action on the 22-torsion of AA has a large image, we prove, under mild additional hypotheses and assuming the finiteness of relevant Shafarevich–Tate groups, that the Hasse principle holds for the associated Kummer varieties. This provides further evidence for the conjecture that the Brauer–Manin obstruction controls rational points on K3 surfaces.
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