The **geometric endomorphism algebra** of an abelian surface $A$ over a field $K$ is the $\Q$-algebra $\End(A_{\overline{K}}) \otimes \Q$, the endomorphism algebra of the base change $A_{\overline{K}}$ of $A$ to an algebraic closure $\overline{K}$ of $K$. This is a special case of the geometric endomorphism algebra of an abelian variety.

When $A$ is an abelian surface over a number field $K$, there are nine possibilities for $\End(A_{\overline K})\otimes\Q$:

- $\Q$;
- a real quadratic field (in which case $A$ has
**real multiplication**, denoted RM); - a quartic CM field (in which case $A$ has
**complex multiplication**, denoted CM); - a non-split quaternion algebra over $\Q$ (in which case $A$ has
**quaternionic multiplication**, denoted QM); - $\Q\times \Q$;
- $F\times \Q$, where $F$ is a quadratic CM field (denoted $\mathrm{CM} \times \Q$);
- $F_1\times F_2$, where $F_1$ and $F_2$ are distinct quadratic CM fields (denoted $\mathrm{CM} \times \mathrm{CM}$);
- $\mathrm{M}_2(\Q)$;
- $\mathrm{M}_2(F)$, where $F$ is a quadratic CM field (denoted $\mathrm{M}_2(\mathrm{CM})$).

The first four cases occur when $A$ is geometrically simple, while the last five cases occur when $A_{\overline K}$ is isogenous to a product of elliptic curves $E_1\times E_2$ over $\overline K$. Which of the last five cases occurs depends on whether $E_1$ and/or $E_2$ have complex multiplication, and whether $E_1$ and $E_2$ are isogenous or not.

**Knowl status:**

- Review status: reviewed
- Last edited by Kiran S. Kedlaya on 2020-10-12 16:19:35

**Referred to by:**

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

- 2023-07-11 11:45:59 by John Jones
- 2020-10-12 16:19:35 by Kiran S. Kedlaya (Reviewed)
- 2020-09-26 16:45:12 by John Voight
- 2020-09-26 16:44:51 by John Voight
- 2020-09-26 16:40:03 by John Voight
- 2020-09-26 16:39:47 by John Voight
- 2019-04-27 15:50:37 by Jennifer Paulhus (Reviewed)
- 2019-04-13 07:20:02 by Andrew Sutherland
- 2019-04-13 07:18:05 by Andrew Sutherland
- 2019-04-12 21:59:46 by Andrew Sutherland
- 2019-04-12 21:59:07 by Andrew Sutherland
- 2019-04-12 21:58:04 by Andrew Sutherland
- 2019-04-12 21:51:11 by Andrew Sutherland

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