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Bobby
Bobby
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The base field is a number field.

It is known that the Shafarevich conjecture implies the Mordell conjecture (Kodaira-Parshin).

Is the converse also true?

Note that both conjectures are now theorems (Faltings).

Edit: To be clear, I'm referring to the Shafarevich conjecture for curves. That is, for any number field $K$, finite set of places $S$ of $K$ and integer $g > 1$, the set of curves over $K$ of genus $g$ with good reduction outside $S$ is finite.

The base field is a number field.

It is known that the Shafarevich conjecture implies the Mordell conjecture (Kodaira-Parshin).

Is the converse also true?

Note that both conjectures are now theorems (Faltings).

The base field is a number field.

It is known that the Shafarevich conjecture implies the Mordell conjecture (Kodaira-Parshin).

Is the converse also true?

Note that both conjectures are now theorems (Faltings).

Edit: To be clear, I'm referring to the Shafarevich conjecture for curves. That is, for any number field $K$, finite set of places $S$ of $K$ and integer $g > 1$, the set of curves over $K$ of genus $g$ with good reduction outside $S$ is finite.

Source Link
Bobby
Bobby
  • 51
  • 2

Does the Mordell conjecture imply the Shafarevich conjecture

The base field is a number field.

It is known that the Shafarevich conjecture implies the Mordell conjecture (Kodaira-Parshin).

Is the converse also true?

Note that both conjectures are now theorems (Faltings).