On the Tate Conjectures for Divisors on a Fibred Variety and on its Generic Scheme Fibre in the Case of Finite Characteristic
We investigate interrelations between the Tate conjecture for divisors on a fibred variety over a finite field and the Tate conjecture for divisors on the generic scheme fibre under the condition that the generic scheme fibre has zero irregularity. Let \(\pi:X\to C\) be a surjective morphism of smoo...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2017-04-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/509 |
| _version_ | 1797878008772558848 |
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| author | Tatyana V. Prokhorova |
| author_facet | Tatyana V. Prokhorova |
| author_sort | Tatyana V. Prokhorova |
| collection | DOAJ |
| description | We investigate interrelations between the Tate conjecture for divisors on a fibred variety over a finite field and the Tate conjecture for divisors on the generic scheme fibre under the condition that the generic scheme fibre has zero irregularity. Let \(\pi:X\to C\) be a surjective morphism of smooth projective varieties over a finite field \(F_q\) of characteristic \(p\), \(C\) is a curve and the generic scheme fibre of \(\pi\) is a smooth variety \(V\) over the field \(k=\kappa(C)\) of rational functions of the curve \(C\), \(\overline k\) is an algebraic closure of the field \(k\), \(k^s\) is its separable closure, \(NS(V)\) is the N\'eron - Severi group of classes of divisors on the variety \(V\) modulo algebraic equivalence, and assume that the following conditions hold: \(H^1(V\otimes\overline k,\mathcal O_{V\otimes\,\overline k})=0,\) \(NS(V)=NS(V\otimes\overline k).\) If, for a prime number \(l\) not dividing \({Card}([NS(V)]_{tors})\) and different from the characteristic of the field \(F_q\), the following relation holds \(NS(V)\otimes\Bbb Q_l\,\,\widetilde{\rightarrow}\,\,[H^2(V\otimes k^{sep},Q_l(1))]^{Gal( k^{sep}/k)} \) \((\)in other words, if the Tate conjecture for divisors on \(V\) holds\()\), then for any prime number \(l\neq charr(F_q)\) the Tate conjecture holds for divisors on \(X\): \(NS(X)\otimes Q_l\,\,\widetilde{\rightarrow} \,\,[H^2(X\otimes\overline F_q,Q_l(1))]^{Gal(\overline F_q/F_q)}.\) In particular, it follows from this result that the Tate conjecture for divisors on an arithmetic model of a \(K3\) surface over a sufficiently large global field of finite characteristic different from 2 holds as well. |
| first_indexed | 2024-04-10T02:25:58Z |
| format | Article |
| id | doaj.art-4efdac09565340df8d398a0eb2170f7a |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2024-04-10T02:25:58Z |
| publishDate | 2017-04-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-4efdac09565340df8d398a0eb2170f7a2023-03-13T08:07:29ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172017-04-0124220521410.18255/1818-1015-2017-2-205-214362On the Tate Conjectures for Divisors on a Fibred Variety and on its Generic Scheme Fibre in the Case of Finite CharacteristicTatyana V. Prokhorova0Владимирский государственный университет им. А.Г. и Н.Г. СтолетовыхWe investigate interrelations between the Tate conjecture for divisors on a fibred variety over a finite field and the Tate conjecture for divisors on the generic scheme fibre under the condition that the generic scheme fibre has zero irregularity. Let \(\pi:X\to C\) be a surjective morphism of smooth projective varieties over a finite field \(F_q\) of characteristic \(p\), \(C\) is a curve and the generic scheme fibre of \(\pi\) is a smooth variety \(V\) over the field \(k=\kappa(C)\) of rational functions of the curve \(C\), \(\overline k\) is an algebraic closure of the field \(k\), \(k^s\) is its separable closure, \(NS(V)\) is the N\'eron - Severi group of classes of divisors on the variety \(V\) modulo algebraic equivalence, and assume that the following conditions hold: \(H^1(V\otimes\overline k,\mathcal O_{V\otimes\,\overline k})=0,\) \(NS(V)=NS(V\otimes\overline k).\) If, for a prime number \(l\) not dividing \({Card}([NS(V)]_{tors})\) and different from the characteristic of the field \(F_q\), the following relation holds \(NS(V)\otimes\Bbb Q_l\,\,\widetilde{\rightarrow}\,\,[H^2(V\otimes k^{sep},Q_l(1))]^{Gal( k^{sep}/k)} \) \((\)in other words, if the Tate conjecture for divisors on \(V\) holds\()\), then for any prime number \(l\neq charr(F_q)\) the Tate conjecture holds for divisors on \(X\): \(NS(X)\otimes Q_l\,\,\widetilde{\rightarrow} \,\,[H^2(X\otimes\overline F_q,Q_l(1))]^{Gal(\overline F_q/F_q)}.\) In particular, it follows from this result that the Tate conjecture for divisors on an arithmetic model of a \(K3\) surface over a sufficiently large global field of finite characteristic different from 2 holds as well.https://www.mais-journal.ru/jour/article/view/509гипотеза тэйтаглобальное полегруппа брауэраарифметическая модельk3 – поверхность |
| spellingShingle | Tatyana V. Prokhorova On the Tate Conjectures for Divisors on a Fibred Variety and on its Generic Scheme Fibre in the Case of Finite Characteristic Моделирование и анализ информационных систем гипотеза тэйта глобальное поле группа брауэра арифметическая модель k3 – поверхность |
| title | On the Tate Conjectures for Divisors on a Fibred Variety and on its Generic Scheme Fibre in the Case of Finite Characteristic |
| title_full | On the Tate Conjectures for Divisors on a Fibred Variety and on its Generic Scheme Fibre in the Case of Finite Characteristic |
| title_fullStr | On the Tate Conjectures for Divisors on a Fibred Variety and on its Generic Scheme Fibre in the Case of Finite Characteristic |
| title_full_unstemmed | On the Tate Conjectures for Divisors on a Fibred Variety and on its Generic Scheme Fibre in the Case of Finite Characteristic |
| title_short | On the Tate Conjectures for Divisors on a Fibred Variety and on its Generic Scheme Fibre in the Case of Finite Characteristic |
| title_sort | on the tate conjectures for divisors on a fibred variety and on its generic scheme fibre in the case of finite characteristic |
| topic | гипотеза тэйта глобальное поле группа брауэра арифметическая модель k3 – поверхность |
| url | https://www.mais-journal.ru/jour/article/view/509 |
| work_keys_str_mv | AT tatyanavprokhorova onthetateconjecturesfordivisorsonafibredvarietyandonitsgenericschemefibreinthecaseoffinitecharacteristic |