An **endomorphism** of an abelian variety $A$ over a field $F$ is a homomorphism $\varphi \colon A \to A$ defined over $F$. The set of endomorphisms of an abelian variety $A$ can be given the structure of a ring in which addition is defined pointwise (using the group operation of $A$) and multiplication is composition; this ring is called the **endomorphism ring** of $A$, denoted $\textrm{End}(A)$.

For endomorphisms defined over an extension of $F$, we instead speak about the geometric endomorphism ring.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Voight on 2020-09-26 17:00:33

**Referred to by:**

- ag.complex_multiplication
- ag.endomorphism_algebra
- ag.geom_endomorphism_ring
- ag.real_endomorphism_algebra
- curve.highergenus.aut.groupalgebradecomp
- ec.endomorphism
- ec.endomorphism_ring
- ec.q.endomorphism_ring
- g2c.277.a.277.1.top
- g2c.jac_end_lattice
- g2c.jac_endomorphisms
- g2c.jacobian
- mf.gl2.history.varieties
- rcs.source.g2c
- rcs.source.st_group

**History:**(expand/hide all)

- 2020-09-26 17:00:33 by John Voight (Reviewed)
- 2020-09-26 17:00:19 by John Voight
- 2020-09-26 16:50:43 by John Voight
- 2020-09-26 15:21:18 by John Voight
- 2020-09-26 15:16:32 by John Voight
- 2020-09-26 15:16:03 by John Voight
- 2020-09-23 11:27:30 by John Voight
- 2018-06-18 12:01:44 by John Cremona (Reviewed)

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