65 results found
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 A, Loughran D, Smeets A, Pseudo-split fibres and arithmetic surjectivity, Annales Scientifiques de l'École Normale Supérieure, ISSN: 0012-9593
Letf:X→Ybe a dominant morphism of smooth, properand geometrically integral varieties over a number fieldk, with geometricallyintegral generic fibre. We give a necessary and sufficient geometric criterionfor the induced mapX(kv)→Y(kv) to be surjective for almost all placesvofk. This generalises a result of Denef which had previously been conjecturedby Colliot-Th ́el`ene, and can be seen as an optimal geometric version of thecelebrated Ax–Kochen theorem.
Skorobogatov AN, Orr M, Finiteness theorems for K3 surfaces and abelian varieties of CM type, Compositio Mathematica, ISSN: 0010-437X
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.
Colliot-Thelene JL, Pal A, Skorobogatov AN, 2016, Pathologies of the Brauer-Manin obstruction, Mathematische Zeitschrift, Vol: 282, Pages: 799-817, ISSN: 1432-1823
Skorobogatov AN, Zarhin YG, 2015, A Finiteness Theorem for the Brauer Group of K3 Surfaces in Odd Characteristic, International Mathematics Research Notices, Vol: 2015, Pages: 11404-11418, ISSN: 1687-0247
Let pp be an odd prime and let kk be a field finitely generated over the finite field with pp elements. For any K3 surface XX over k,k, we prove that the cokernel of the natural map Br(k)→Br(X)Br(k)→Br(X) is finite modulo the pp-primary torsion subgroup.
Harpaz Y, Skorobogatov AN, Wittenberg O, 2014, The Hardy-Littlewood conjecture and rational points, COMPOSITIO MATHEMATICA, Vol: 150, Pages: 2095-2111, ISSN: 0010-437X
Ieronymou E, Skorobogatov AN, 2014, Odd order Brauer-Manin obstruction on diagonal quartic surfaces, Advances in Mathematics, Vol: 270, Pages: 181-205, ISSN: 1090-2082
We determine the odd order torsion subgroup of the Brauer group of diagonal quartic surfaces over the field of rational numbers. We show that a non-constant Brauer element of odd order always obstructs weak approximation but never the Hasse principle.
Harpaz Y, Skorobogatov AN, 2014, Singular curves and the étale Brauer-Manin obstruction for surfaces, Annales scientifiques de l'École normale supérieure, Vol: 47, Pages: 765-778, ISSN: 0012-9593
We give an elementary construction of a smooth and projective surface over an arbitrary number field k that is a counterexample to the Hasse principle but has infinite étale Brauer-Manin set. Our surface has a surjective morphism to a curve with exactly one k-point such that the unique k-fibre is geometrically a union of projective lines with an adelic point and the trivial Brauer group, but no k-point
Browning TD, Matthiesen L, Skorobogatov AN, 2014, Rational points on pencils of conics and quadrics with many degenerate fibres, Annals of Mathematics, Vol: 180, Pages: 381-402, ISSN: 1939-8980
For any pencil of conics or higher-dimensional quadrics over Q, with all degenerate fibres defined over Q, we show that the Brauer–Manin obstruction controls weak approximation. The proof is based on the Hasse principle and weak approximation for some special intersections of quadrics over Q, which is a consequence of recent advances in additive combinatorics.
Skorobogatov AN, Zarhin YG, 2014, The Brauer group and the Brauer-Manin set of products of varieties, Journal of the European Mathematical Society, Vol: 16, Pages: 749-769, ISSN: 1435-9863
Let XX and YY be smooth and projective varieties over a field kk finitely generated over \Q\Q, and let \ovX\ovX and \ovY\ovY be the varieties over an algebraic closure of kk obtained from XX and YY, respectively, by extension of the ground field. We show that the Galois invariant subgroup of \Br(\ovX)⊕\Br(\ovY)\Br(\ovX)⊕\Br(\ovY) has finite index in the Galois invariant subgroup of \Br(\ovX×\ovY)\Br(\ovX×\ovY). This implies that the cokernel of the natural map \Br(X)⊕\Br(Y)→\Br(X×Y)\Br(X)⊕\Br(Y)→\Br(X×Y) is finite when kk is a number field. In this case we prove that the Brauer–Manin set of the product of varieties is the product of their Brauer–Manin sets.
Schindler D, Skorobogatov A, 2014, Norms as products of linear polynomials, Journal of the London Mathematical Society-Second Series, Vol: 89, Pages: 559-580, ISSN: 1469-7750
Let FF be a number field, and let F⊂KF⊂K be a field extension of degree nn. Suppose that we are given 2r2r sufficiently general linear polynomials in rr variables over FF. Let XX be the variety over FF such that the FF-points of XX bijectively correspond to the representations of the product of these polynomials by a norm from KK to FF. Combining the circle method with descent we prove that the Brauer–Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of XX.
Colliot-Thelene J-L, Skorobogatov AN, 2013, Galois descent on the Brauer group, JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, Vol: 682, Pages: 141-165, ISSN: 0075-4102
Hausen J, Schlank TM, Skorobogatov AN, et al., 2013, Torsors, Étale Homotopy and Applications to Rational Points, Publisher: Cambridge University Press, ISBN: 9781107616127
Lecture notes and research articles on the use of torsors and étale homotopy in algebraic and arithmetic geometry.
Colliot-Thelene J-L, Skorobogatov AN, 2013, GOOD REDUCTION OF THE BRAUER-MANIN OBSTRUCTION, TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 365, Pages: 579-590, ISSN: 0002-9947
Skorobogatov AN, Zarhin YG, 2012, The Brauer group of Kummer surfaces and torsion of elliptic curves, JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, Vol: 666, Pages: 115-140, ISSN: 0075-4102
Ieronymou E, Skorobogatov AN, Zarhin YG, 2011, On the Brauer group of diagonal quartic surfaces, JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, Vol: 83, Pages: 659-672, ISSN: 0024-6107
Serganova VV, Skorobogatov AN, 2011, ADJOINT REPRESENTATION OF E-8 AND DEL PEZZO SURFACES OF DEGREE 1, ANNALES DE L INSTITUT FOURIER, Vol: 61, Pages: 2337-2360, ISSN: 0373-0956
Serganova VV, Skorobogatov AN, 2010, ON THE EQUATIONS FOR UNIVERSAL TORSORS OVER DEL PEZZO SURFACES, JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU, Vol: 9, Pages: 203-223, ISSN: 1474-7480
Skorobogatov AN, 2009, Automorphisms and forms of toric quotients of homogeneous spaces, SBORNIK MATHEMATICS, Vol: 200, Pages: 1521-1536, ISSN: 1064-5616
Skorobogatov A, 2009, Descent obstruction is equivalent to ,tale Brauer-Manin obstruction, MATHEMATISCHE ANNALEN, Vol: 344, Pages: 501-510, ISSN: 0025-5831
Skorobogatov AN, Zarhin YG, 2008, A FINITENESS THEOREM FOR THE BRAUER GROUP OF ABELIAN VARIETIES AND K3 SURFACES, JOURNAL OF ALGEBRAIC GEOMETRY, Vol: 17, Pages: 481-502, ISSN: 1056-3911
Let k be a field finitely generated over the field of rational numbers, and Br(k) the Brauer group of k. For an algebraic variety X over k we consider the cohomological Brauer-Grothendieck group Br(X). We prove that the quotient of Br(X) by the image of Br(k) is finite if X is a K3 surface. When X is an abelian variety over k, and X is the variety over an algebraic closure (k) over bar of k obtained from X by the extension of the ground field, we prove that the image of Br(X) in Br(X) is finite.
Colliot-Thélène JL, Borovoi M, Skorobogatov A, 2008, The elementary obstruction and homogeneous spaces, Duke Mathematical Journal, Vol: 141, Pages: 321-364
Skorobogatov AN, Zarhin YG, 2008, A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces., J. Alg. Geom
Skorobogatov A, Serganova V, Del Pezzo surfaces and representation theory., J. Algebra and Number Theory
Skorobogatov AN, 2007, On the elementary obstruction to the existence of rational points, MATHEMATICAL NOTES, Vol: 81, Pages: 97-107, ISSN: 0001-4346
Skorobogatov A, Serganova V, 2007, Del Pezzo surfaces and representation theory., J. Algebra and Number Theory, Vol: 1, Pages: 393-419
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