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I have a question about proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite. Could you please tell (correct) me any strange parts about my understanding of the outline of the proof ?

My understanding (outline)

"It is difficult to prove elementaryly that $E (K) / E_0 (K)$ is finite. Therefore, we use the Neron model. The"The elliptic curve $E$ on the local field $K $($R$ is an ring of integers) is the Neron model ( There is a group scheme on $SpecR$ that has $E / K$ as a generic fiber), and if it is $ε$, then $E (K) / E_0 (K)$ is isomorphic to $ε (R) / ε_0 (R)$ as a scheme. Furthermore, since $K$ is a Hansel field, $ε (R) / ε_0 (R)$ is isomorphic to $ε'(R) / ε'_0 (R)$ (where $ε'$ is a special fiber of $ε$ and $ε'_0$ is an connected component of $ε'$ which contains identity element). Since the algebraic group $ε'_0$ is a connected component containing the identity element of the algebraic group ε', by using the algebraic group theorem “If $G$ is an algebraic group, then the connected component of $G$ including the identity element is a subgroup with a finite index ", we can conclude that it is a subgroup of a finite index.

The algebraic group theorem cannot be used for the original $E (K) / E_0 (K)$. This is because $E_0 (K)$ is not connected unless $E_0(K)＝E (K)$. Therefore, the algebraic group theorem can be applied via the Neron model and its special fiber. "

I have a question about proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite. Could you please tell (correct) me any strange parts about my understanding of the outline of the proof ?

My understanding (outline)

"It is difficult to prove elementaryly that $E (K) / E_0 (K)$ is finite. Therefore, we use the Neron model. The elliptic curve $E$ on the local field $K $($R$ is an ring of integers) is the Neron model ( There is a group scheme on $SpecR$ that has $E / K$ as a generic fiber), and if it is $ε$, then $E (K) / E_0 (K)$ is isomorphic to $ε (R) / ε_0 (R)$ as a scheme. Furthermore, since $K$ is a Hansel field, $ε (R) / ε_0 (R)$ is isomorphic to $ε'(R) / ε'_0 (R)$ (where $ε'$ is a special fiber of $ε$ and $ε'_0$ is an connected component of $ε'$ which contains identity element). Since the algebraic group $ε'_0$ is a connected component containing the identity element of the algebraic group ε', by using the algebraic group theorem “If $G$ is an algebraic group, then the connected component of $G$ including the identity element is a subgroup with a finite index ", we can conclude that it is a subgroup of a finite index.

The algebraic group theorem cannot be used for the original $E (K) / E_0 (K)$. This is because $E_0 (K)$ is not connected unless $E_0(K)＝E (K)$. Therefore, the algebraic group theorem can be applied via the Neron model and its special fiber. "

I have a question about proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite. Could you please tell (correct) me any strange parts about my understanding of the outline of the proof ?

My understanding (outline)

"The elliptic curve $E$ on the local field $K $($R$ is an ring of integers) is the Neron model ( There is a group scheme on $SpecR$ that has $E / K$ as a generic fiber), and if it is $ε$, then $E (K) / E_0 (K)$ is isomorphic to $ε (R) / ε_0 (R)$ as a scheme. Furthermore, since $K$ is a Hansel field, $ε (R) / ε_0 (R)$ is isomorphic to $ε'(R) / ε'_0 (R)$ (where $ε'$ is a special fiber of $ε$ and $ε'_0$ is an connected component of $ε'$ which contains identity element). Since the algebraic group $ε'_0$ is a connected component containing the identity element of the algebraic group ε', by using the algebraic group theorem “If $G$ is an algebraic group, then the connected component of $G$ including the identity element is a subgroup with a finite index ", we can conclude that it is a subgroup of a finite index.

The algebraic group theorem cannot be used for the original $E (K) / E_0 (K)$. This is because $E_0 (K)$ is not connected unless $E_0(K)＝E (K)$. Therefore, the algebraic group theorem can be applied via the Neron model and its special fiber. "

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Duality
Duality
  • 1.5k
  • 7
  • 13

Outline of the proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite

I have a question about proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite. Could you please tell (correct) me any strange parts about my understanding of the outline of the proof ?

My understanding (outline)

"It is difficult to prove elementaryly that $E (K) / E_0 (K)$ is finite. Therefore, we use the Neron model. The elliptic curve $E$ on the local field $K $($R$ is an ring of integers) is the Neron model ( There is a group scheme on $SpecR$ that has $E / K$ as a generic fiber), and if it is $ε$, then $E (K) / E_0 (K)$ is isomorphic to $ε (R) / ε_0 (R)$ as a scheme. Furthermore, since $K$ is a Hansel field, $ε (R) / ε_0 (R)$ is isomorphic to $ε'(R) / ε'_0 (R)$ (where $ε'$ is a special fiber of $ε$ and $ε'_0$ is an connected component of $ε'$ which contains identity element). Since the algebraic group $ε'_0$ is a connected component containing the identity element of the algebraic group ε', by using the algebraic group theorem “If $G$ is an algebraic group, then the connected component of $G$ including the identity element is a subgroup with a finite index ", we can conclude that it is a subgroup of a finite index.

The algebraic group theorem cannot be used for the original $E (K) / E_0 (K)$. This is because $E_0 (K)$ is not connected unless $E_0(K)＝E (K)$. Therefore, the algebraic group theorem can be applied via the Neron model and its special fiber. "