Skip to main content
MathOverflow

Return to Question

deleted 95 characters in body
Source Link
user94490
user94490

For elliptic curves, one may easily compute Mumford-Tate groups; there are just two cases:

  1. $E$ has no complex multiplication, and the Mumford-Tate group of $E$ is $GL_2$

  2. $E$ has complex multiplication by a field $k$ and the Mumford-Tate group $E$ is a torus (of dimension two) in $GL_2$ induced by $k$.

In any case, the list of possible dimensions of Mumford-Tate groups of elliptic curves is $\{2,4\}$.

Is it possible to explicitly classify all the Mumford-Tate groups of abelian surfaces? What about special cases, such as abelian surfaces $A$ such that $End^0(A)$ is real quadratic. What is the list of possible dimensions of Mumford-Tate groups in this case?

For elliptic curves, one may easily compute Mumford-Tate groups; there are just two cases:

  1. $E$ has no complex multiplication, and the Mumford-Tate group of $E$ is $GL_2$

  2. $E$ has complex multiplication by a field $k$ and the Mumford-Tate group $E$ is a torus (of dimension two) in $GL_2$ induced by $k$.

In any case, the list of possible dimensions of Mumford-Tate groups of elliptic curves is $\{2,4\}$.

Is it possible to explicitly classify all the Mumford-Tate groups of abelian surfaces? What about special cases, such as abelian surfaces $A$ such that $End^0(A)$ is real quadratic. What is the list of possible dimensions of Mumford-Tate groups in this case?

For elliptic curves, one may easily compute Mumford-Tate groups; there are just two cases:

  1. $E$ has no complex multiplication, and the Mumford-Tate group of $E$ is $GL_2$

  2. $E$ has complex multiplication by a field $k$ and the Mumford-Tate group $E$ is a torus (of dimension two) in $GL_2$ induced by $k$.

In any case, the list of possible dimensions of Mumford-Tate groups of elliptic curves is $\{2,4\}$.

Is it possible to explicitly classify all the Mumford-Tate groups of abelian surfaces? What is the list of possible dimensions of Mumford-Tate groups in this case?

Source Link
user94490
user94490

Mumford-Tate groups of abelian surfaces

For elliptic curves, one may easily compute Mumford-Tate groups; there are just two cases:

  1. $E$ has no complex multiplication, and the Mumford-Tate group of $E$ is $GL_2$

  2. $E$ has complex multiplication by a field $k$ and the Mumford-Tate group $E$ is a torus (of dimension two) in $GL_2$ induced by $k$.

In any case, the list of possible dimensions of Mumford-Tate groups of elliptic curves is $\{2,4\}$.

Is it possible to explicitly classify all the Mumford-Tate groups of abelian surfaces? What about special cases, such as abelian surfaces $A$ such that $End^0(A)$ is real quadratic. What is the list of possible dimensions of Mumford-Tate groups in this case?