Isogeny

In algebraic geometry , a branch of mathematics , one calls a homomorphism of Abelian varieties and an isogeny if it is surjective and has a finite kernel . If there is isogeny , the Abelian varieties are called and isogenic . ${\displaystyle \phi \colon A\rightarrow B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \phi }$ ${\displaystyle \phi \colon A\rightarrow B}$ ${\displaystyle A}$ ${\displaystyle B}$

definition

If and are Abelian varieties, then the following statements about a homomorphism are equivalent: ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \phi \colon A\rightarrow B}$

• ${\displaystyle \phi }$is an isogeny, that is, is surjective and the core of is finite. ${\displaystyle \phi }$ ${\displaystyle \phi }$
• ${\displaystyle A}$and have the same dimension and are surjective. ${\displaystyle B}$ ${\displaystyle \phi }$
• ${\displaystyle A}$and have the same dimension and the core of is finite. ${\displaystyle B}$ ${\displaystyle \phi }$

Is a (and therefore any) of these conditions are met, it is called and isogenic. ${\displaystyle A}$ ${\displaystyle B}$

The concept of an isogeny of Abelian varieties dealt with in this article can be generalized to the concept of an isogeny of group schemes .

Individual evidence

1. ↑ James Milne : Abelian Varieties . Course Notes, version 2.0, 2008, Proposition 7.1. (English)

literature

• Serge Lang : Abelian Varieties . Springer Verlag, New York 1983, ISBN 3-540-90875-7 (English).
• David Mumford : Abelian Varieties . Oxford University Press, London 1974, ISBN 0-19-560528-4 (English).