Skip to main content
MathOverflow

Return to Question

+ top level tag (nt. & ag.) + correction due to Daniel Loughran & minor latex edits
Source Link
Myshkin
Myshkin
  • 17.6k
  • 5
  • 71
  • 137

Possible groups of K-rational ptspoints for elliptic curves over arbitrary fields

It is known that the group $\ C(\Bbb R)$$C(\Bbb R)$ has at most two connected components, and the connected component of the identity is isomorphic to U(1)$U(1)$ as a topological group (trivially) and $\ C(\Bbb Q)$$C(\Bbb Q)$ is isomorphic to $\Bbb Z^r\times E(\Bbb Q)$, where all possible $\ E(\Bbb Q)$$E(\Bbb Q)$ are known. 

Are there similar results for other fields? I am especially interested in the case K is a number field, or the p-adic numbers (both $\Bbb Z_p$ and $\Bbb Q_p$)

Possible groups of K-rational pts for elliptic curves over arbitrary fields

It is known that the group $\ C(\Bbb R)$ is isomorphic to U(1) (trivially) and $\ C(\Bbb Q)$ is isomorphic to $\Bbb Z^r\times E(\Bbb Q)$, where all possible $\ E(\Bbb Q)$ are known. Are there similar results for other fields? I am especially interested in the case K is a number field, or the p-adic numbers (both $\Bbb Z_p$ and $\Bbb Q_p$)

Possible groups of K-rational points for elliptic curves over arbitrary fields

It is known that the group $C(\Bbb R)$ has at most two connected components, and the connected component of the identity is isomorphic to $U(1)$ as a topological group (trivially) and $C(\Bbb Q)$ is isomorphic to $\Bbb Z^r\times E(\Bbb Q)$, where all possible $E(\Bbb Q)$ are known. 

Are there similar results for other fields? I am especially interested in the case K is a number field, or the p-adic numbers (both $\Bbb Z_p$ and $\Bbb Q_p$)

Source Link
FusRoDah
FusRoDah
  • 3.7k
  • 1
  • 16
  • 36

Possible groups of K-rational pts for elliptic curves over arbitrary fields

It is known that the group $\ C(\Bbb R)$ is isomorphic to U(1) (trivially) and $\ C(\Bbb Q)$ is isomorphic to $\Bbb Z^r\times E(\Bbb Q)$, where all possible $\ E(\Bbb Q)$ are known. Are there similar results for other fields? I am especially interested in the case K is a number field, or the p-adic numbers (both $\Bbb Z_p$ and $\Bbb Q_p$)