### Abstract

An analog of the Tate hypothesis on homomorphisms of Abelian varieties is proved, in which points of sufficiently large prime order figure in place of the Tate modules. As is the case with the Tate hypothesis, this assertion follows formally from a finiteness hypothesis for isogenies of Abelian varieties, which is proved in characteristic p > 2 and for finite fields. The same methods are used to prove the finiteness of the set of Abelian varieties of a given dimension over a finite field.

Original language | English (US) |
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Pages (from-to) | 415-419 |

Number of pages | 5 |

Journal | Mathematical Notes of the Academy of Sciences of the USSR |

Volume | 21 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 1977 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

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**Endomorphisms of abelian varieties and points of finite order in characteristic p.** / Zarkhin, Yuriy G.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Endomorphisms of abelian varieties and points of finite order in characteristic p

AU - Zarkhin, Yuriy G.

PY - 1977/6/1

Y1 - 1977/6/1

N2 - An analog of the Tate hypothesis on homomorphisms of Abelian varieties is proved, in which points of sufficiently large prime order figure in place of the Tate modules. As is the case with the Tate hypothesis, this assertion follows formally from a finiteness hypothesis for isogenies of Abelian varieties, which is proved in characteristic p > 2 and for finite fields. The same methods are used to prove the finiteness of the set of Abelian varieties of a given dimension over a finite field.

AB - An analog of the Tate hypothesis on homomorphisms of Abelian varieties is proved, in which points of sufficiently large prime order figure in place of the Tate modules. As is the case with the Tate hypothesis, this assertion follows formally from a finiteness hypothesis for isogenies of Abelian varieties, which is proved in characteristic p > 2 and for finite fields. The same methods are used to prove the finiteness of the set of Abelian varieties of a given dimension over a finite field.

UR - http://www.scopus.com/inward/record.url?scp=0000057684&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000057684&partnerID=8YFLogxK

U2 - 10.1007/BF01410167

DO - 10.1007/BF01410167

M3 - Article

VL - 21

SP - 415

EP - 419

JO - Mathematical Notes

JF - Mathematical Notes

SN - 0001-4346

IS - 6

ER -