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The geometric endomorphism algebra of an abelian variety $A/k$ is the endomorphism algebra of $A_{\overline k}$.

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

  1. $\Q$;
  2. a real quadratic field (in which case $A$ has real multiplication, denoted RM);
  3. a quartic CM field (in which case $A$ has complex multiplication, denoted CM);
  4. a non-split quaternion algebra over $\Q$ (in which case $A$ has quaternionic multiplication, denoted QM);
  5. $\Q\times \Q$;
  6. $F\times \Q$, where $F$ is a quadratic CM field (denoted $\mathrm{CM} \times \Q$);
  7. $F_1\times F_2$, where $F_1$ and $F_2$ are distinct quadratic CM fields (denoted $\mathrm{CM} \times \mathrm{CM}$);
  8. $\mathrm{M}_2(\Q)$;
  9. $\mathrm{M}_2(F)$, 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 Jennifer Paulhus on 2019-04-27 15:50:37
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