Yuri Zarhin - Academia.edu (original) (raw)

Papers by Yuri Zarhin

Transactions of the American Mathematical Society, Dec 3, 2015

arXiv (Cornell University), Jul 13, 2021

We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finit... more We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.

Mathematical Research Letters, 2007

A celebrated theorem of Bogomolov asserts that the-adic Lie algebra attached to the Galois action... more A celebrated theorem of Bogomolov asserts that the-adic Lie algebra attached to the Galois action on the Tate module of an abelian variety over a number field contains all homotheties. This is not the case in characteristic p: a "counterexample" is provided by an ordinary elliptic curve defined over a finite field. In this note we discuss (and explicitly construct) more interesting examples of "non-constant" absolutely simple abelian varieties (without homotheties) over global fields in characteristic p.

Transformation Groups, Dec 17, 2014

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds ... more We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of algebraic surfaces. This gives a positive answer to a question of

Transformation Groups, Aug 11, 2018

Let W be a quasiprojective variety over an algebraically closed field of characteristic zero. Ass... more Let W be a quasiprojective variety over an algebraically closed field of characteristic zero. Assume that W is birational to a product of a smooth projective variety A and the projective line. We prove that if A contains no rational curves then the automorphism group G := Aut(W) of W is Jordan. That means that there is a positive integer J = J(W) such that every finite subgroup B of G contains a commutative subgroup A such that A is normal in B and the index [B : A] ≤ J .

Contemporary mathematics, 2019

Mathematical Research Letters, 2010

Proceedings of the Steklov Institute of Mathematics, Nov 1, 2019

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does n... more We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does not hold for the group of bimeromorphic automorphisms of a product of the complex projective line and a complex torus of positive algebraic dimension.

Pure and Applied Mathematics Quarterly, 2017

We study Kummer varieties attached to 2-coverings of abelian varieties of arbitrary dimension. Ov... more We study Kummer varieties attached to 2-coverings of abelian varieties of arbitrary dimension. Over a number field we show that the subgroup of odd order elements of the Brauer group does not obstruct the Hasse principle. Sufficient conditions for the triviality of the Brauer group are given, which allow us to give an example of a Kummer K3 surface of geometric Picard rank 17 over the rationals with trivial Brauer group. We establish the nonemptyness of the Brauer-Manin set of everywhere locally soluble Kummer varieties attached to 2-coverings of products of hyperelliptic Jacobians with large Galois action on 2-torsion.

Algebraic geometry, Mar 15, 2017

A group G is called Jordan if there is a positive integer J = J G such that every finite subgroup... more A group G is called Jordan if there is a positive integer J = J G such that every finite subgroup B of G contains a commutative subgroup A ⊂ B such that A is normal in B and the index [B : A] is at most J (V. L. Popov). In this paper, we deal with Jordan properties of the groups Bir(X) of birational automorphisms of irreducible smooth projective varieties X over an algebraically closed field of characteristic zero. It is known (Yu. Prokhorov and C. Shramov) that Bir(X) is Jordan if X is non-uniruled. On the other hand, the second-named author proved that Bir(X) is not Jordan if X is birational to a product of the projective line P 1 and a positive-dimensional abelian variety. We prove that Bir(X) is Jordan if (the uniruled variety) X is a conic bundle over a non-uniruled variety Y but is not birational to Y × P 1. (Such a conic bundle exists if and only if dim(Y) 2.) When Y is an abelian surface, this Jordan property result gives an answer to a question of Prokhorov and Shramov.

arXiv (Cornell University), Feb 7, 2017

We give a simple proof of the well-known divisibility by 2 condition for rational points on ellip... more We give a simple proof of the well-known divisibility by 2 condition for rational points on elliptic curves with rational 2-torsion. As an application of the explicit division by 2 n formulas obtained in Sec.2, we construct versal families of elliptic curves containing points of orders 4, 5, 6, and 8 from which we obtain an explicit description of elliptic curves over certain finite fields Fq with a prescribed (small) group E(Fq). In the last two sections we study 3and 5-torsion.

arXiv (Cornell University), Mar 12, 2021

This paper is a continuation of our article [BZ20]. The notion of a poor complex compact manifold... more This paper is a continuation of our article [BZ20]. The notion of a poor complex compact manifold was introduced there and the group Aut(X) for a P 1 −bundle over such a manifold was proven to be very Jordan. We call a group G very Jordan if it contains a normal abelian subgroup G 0 such that the orders of finite subgroups of the quotient G/G 0 are bounded by a constant depending on G only. In this paper we provide examples of infinite families of poor manifolds of any complex dimension, namely simple tori of algebraic dimension zero. Then we consider a non-trivial holomorphic P 1 −bundle (X, p, Y) over a non-uniruled complex compact Kähler manifold Y. We prove that Aut(X) is very Jordan provided some additional conditions on the set of sections of p are met. Applications to P 1-bundles over non-algebraic complex tori are given.

arXiv (Cornell University), Nov 10, 2007

arXiv (Cornell University), Aug 6, 2005

arXiv (Cornell University), Oct 13, 2019

We call a group G very Jordan if it contains a normal abelian subgroup G 0 such that the orders o... more We call a group G very Jordan if it contains a normal abelian subgroup G 0 such that the orders of finite subgroups of the quotient G/G 0 are bounded by a constant depending on G only. Let Y be a complex torus of algebraic dimension 0. We prove that if X is a non-trivial holomorphic P 1 −bundle over Y then the group Bim(X) of its bimeromorphic automorphisms is very Jordan (contrary to the case when Y has positive algebraic dimension). This assertion remains true if Y is any connected compact complex Kähler manifold of algebraic dimension 0 without rational curves or analytic subsets of codimension 1.

arXiv (Cornell University), Dec 22, 2010

We study analogues of Tate's conjecture on homomorphisms for abelian varieties when the ground fi... more We study analogues of Tate's conjecture on homomorphisms for abelian varieties when the ground field is finitely generated over an algebraic closure of a finite field. Our results cover the case of abelian varieties without nontrivial endomorphisms.

European journal of mathematics, May 5, 2015

We compute the Galois groups for a certain class of polynomials over the the field of rational nu... more We compute the Galois groups for a certain class of polynomials over the the field of rational numbers that was introduced by Shigefumi Mori and study the monodromy of corresponding hyperelliptic jacobians. Keywords Abelian varieties • Hyperelliptic curves • Tate modules • Galois groups Mathematics Subject Classification 14H40 • 14K05 • 11G30 • 11G10 1 Mori polynomials, their reductions and Galois groups We write Z, Q and C for the ring of integers, the field of rational numbers and the field of complex numbers respectively. If a and b are nonzero integers then we write (a, b) for its (positive) greatest common divisor. If is a prime then F , Z and Q stand for the prime finite field of characteristic , the ring of-adic integers and the field of-adic numbers respectively.

Proceedings of the Edinburgh Mathematical Society, Dec 17, 2013

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does n... more We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does not hold for the groups of birational automorphisms of products of an elliptic curve and the projective line. This gives a negative answer to a question of V.

Mathematical Research Letters, 2000

arXiv (Cornell University), Jan 29, 2014

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds ... more We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of elliptic ruled surfaces. This gives a positive answer to a question of Vladimir

Transactions of the American Mathematical Society, Dec 3, 2015

arXiv (Cornell University), Jul 13, 2021

We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finit... more We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.

Mathematical Research Letters, 2007

A celebrated theorem of Bogomolov asserts that the-adic Lie algebra attached to the Galois action... more A celebrated theorem of Bogomolov asserts that the-adic Lie algebra attached to the Galois action on the Tate module of an abelian variety over a number field contains all homotheties. This is not the case in characteristic p: a "counterexample" is provided by an ordinary elliptic curve defined over a finite field. In this note we discuss (and explicitly construct) more interesting examples of "non-constant" absolutely simple abelian varieties (without homotheties) over global fields in characteristic p.

Transformation Groups, Dec 17, 2014

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds ... more We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of algebraic surfaces. This gives a positive answer to a question of

Transformation Groups, Aug 11, 2018

Let W be a quasiprojective variety over an algebraically closed field of characteristic zero. Ass... more Let W be a quasiprojective variety over an algebraically closed field of characteristic zero. Assume that W is birational to a product of a smooth projective variety A and the projective line. We prove that if A contains no rational curves then the automorphism group G := Aut(W) of W is Jordan. That means that there is a positive integer J = J(W) such that every finite subgroup B of G contains a commutative subgroup A such that A is normal in B and the index [B : A] ≤ J .

Contemporary mathematics, 2019

Mathematical Research Letters, 2010

Proceedings of the Steklov Institute of Mathematics, Nov 1, 2019

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does n... more We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does not hold for the group of bimeromorphic automorphisms of a product of the complex projective line and a complex torus of positive algebraic dimension.

Pure and Applied Mathematics Quarterly, 2017

We study Kummer varieties attached to 2-coverings of abelian varieties of arbitrary dimension. Ov... more We study Kummer varieties attached to 2-coverings of abelian varieties of arbitrary dimension. Over a number field we show that the subgroup of odd order elements of the Brauer group does not obstruct the Hasse principle. Sufficient conditions for the triviality of the Brauer group are given, which allow us to give an example of a Kummer K3 surface of geometric Picard rank 17 over the rationals with trivial Brauer group. We establish the nonemptyness of the Brauer-Manin set of everywhere locally soluble Kummer varieties attached to 2-coverings of products of hyperelliptic Jacobians with large Galois action on 2-torsion.

Algebraic geometry, Mar 15, 2017

A group G is called Jordan if there is a positive integer J = J G such that every finite subgroup... more A group G is called Jordan if there is a positive integer J = J G such that every finite subgroup B of G contains a commutative subgroup A ⊂ B such that A is normal in B and the index [B : A] is at most J (V. L. Popov). In this paper, we deal with Jordan properties of the groups Bir(X) of birational automorphisms of irreducible smooth projective varieties X over an algebraically closed field of characteristic zero. It is known (Yu. Prokhorov and C. Shramov) that Bir(X) is Jordan if X is non-uniruled. On the other hand, the second-named author proved that Bir(X) is not Jordan if X is birational to a product of the projective line P 1 and a positive-dimensional abelian variety. We prove that Bir(X) is Jordan if (the uniruled variety) X is a conic bundle over a non-uniruled variety Y but is not birational to Y × P 1. (Such a conic bundle exists if and only if dim(Y) 2.) When Y is an abelian surface, this Jordan property result gives an answer to a question of Prokhorov and Shramov.

arXiv (Cornell University), Feb 7, 2017

We give a simple proof of the well-known divisibility by 2 condition for rational points on ellip... more We give a simple proof of the well-known divisibility by 2 condition for rational points on elliptic curves with rational 2-torsion. As an application of the explicit division by 2 n formulas obtained in Sec.2, we construct versal families of elliptic curves containing points of orders 4, 5, 6, and 8 from which we obtain an explicit description of elliptic curves over certain finite fields Fq with a prescribed (small) group E(Fq). In the last two sections we study 3and 5-torsion.

arXiv (Cornell University), Mar 12, 2021

This paper is a continuation of our article [BZ20]. The notion of a poor complex compact manifold... more This paper is a continuation of our article [BZ20]. The notion of a poor complex compact manifold was introduced there and the group Aut(X) for a P 1 −bundle over such a manifold was proven to be very Jordan. We call a group G very Jordan if it contains a normal abelian subgroup G 0 such that the orders of finite subgroups of the quotient G/G 0 are bounded by a constant depending on G only. In this paper we provide examples of infinite families of poor manifolds of any complex dimension, namely simple tori of algebraic dimension zero. Then we consider a non-trivial holomorphic P 1 −bundle (X, p, Y) over a non-uniruled complex compact Kähler manifold Y. We prove that Aut(X) is very Jordan provided some additional conditions on the set of sections of p are met. Applications to P 1-bundles over non-algebraic complex tori are given.

arXiv (Cornell University), Nov 10, 2007

arXiv (Cornell University), Aug 6, 2005

arXiv (Cornell University), Oct 13, 2019

We call a group G very Jordan if it contains a normal abelian subgroup G 0 such that the orders o... more We call a group G very Jordan if it contains a normal abelian subgroup G 0 such that the orders of finite subgroups of the quotient G/G 0 are bounded by a constant depending on G only. Let Y be a complex torus of algebraic dimension 0. We prove that if X is a non-trivial holomorphic P 1 −bundle over Y then the group Bim(X) of its bimeromorphic automorphisms is very Jordan (contrary to the case when Y has positive algebraic dimension). This assertion remains true if Y is any connected compact complex Kähler manifold of algebraic dimension 0 without rational curves or analytic subsets of codimension 1.

arXiv (Cornell University), Dec 22, 2010

We study analogues of Tate's conjecture on homomorphisms for abelian varieties when the ground fi... more We study analogues of Tate's conjecture on homomorphisms for abelian varieties when the ground field is finitely generated over an algebraic closure of a finite field. Our results cover the case of abelian varieties without nontrivial endomorphisms.

European journal of mathematics, May 5, 2015

We compute the Galois groups for a certain class of polynomials over the the field of rational nu... more We compute the Galois groups for a certain class of polynomials over the the field of rational numbers that was introduced by Shigefumi Mori and study the monodromy of corresponding hyperelliptic jacobians. Keywords Abelian varieties • Hyperelliptic curves • Tate modules • Galois groups Mathematics Subject Classification 14H40 • 14K05 • 11G30 • 11G10 1 Mori polynomials, their reductions and Galois groups We write Z, Q and C for the ring of integers, the field of rational numbers and the field of complex numbers respectively. If a and b are nonzero integers then we write (a, b) for its (positive) greatest common divisor. If is a prime then F , Z and Q stand for the prime finite field of characteristic , the ring of-adic integers and the field of-adic numbers respectively.

Proceedings of the Edinburgh Mathematical Society, Dec 17, 2013

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does n... more We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does not hold for the groups of birational automorphisms of products of an elliptic curve and the projective line. This gives a negative answer to a question of V.

Mathematical Research Letters, 2000

arXiv (Cornell University), Jan 29, 2014

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds ... more We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of elliptic ruled surfaces. This gives a positive answer to a question of Vladimir