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I only know this material in a vague and impressionistic way, but I think that it the fact that $0$ goes to the identity on the elliptic curve follows from Eichler-Shimura theory.

If $E/\mathbb{Q}$ is parametrized by a modular curve, then it's a quotient of the Jacobian of the modular curve. In the composite map $$X_{0}(N) \to J_{0}(N) \to E$$ each arrow sends $0$ to $0$.

[Edit: Looking at this questionthis question I see that apparently 0 need not go to the identity in general. So now I'm puzzled. I think that my confusion might come from not knowing the definition of "0" on the modular curve in this context, which I was taking to be the preimage of the identity on the Jacobian.]

I only know this material in a vague and impressionistic way, but I think that it the fact that $0$ goes to the identity on the elliptic curve follows from Eichler-Shimura theory.

If $E/\mathbb{Q}$ is parametrized by a modular curve, then it's a quotient of the Jacobian of the modular curve. In the composite map $$X_{0}(N) \to J_{0}(N) \to E$$ each arrow sends $0$ to $0$.

[Edit: Looking at this question I see that apparently 0 need not go to the identity in general. So now I'm puzzled. I think that my confusion might come from not knowing the definition of "0" on the modular curve in this context, which I was taking to be the preimage of the identity on the Jacobian.]

I only know this material in a vague and impressionistic way, but I think that it the fact that $0$ goes to the identity on the elliptic curve follows from Eichler-Shimura theory.

If $E/\mathbb{Q}$ is parametrized by a modular curve, then it's a quotient of the Jacobian of the modular curve. In the composite map $$X_{0}(N) \to J_{0}(N) \to E$$ each arrow sends $0$ to $0$.

[Edit: Looking at this question I see that apparently 0 need not go to the identity in general. So now I'm puzzled. I think that my confusion might come from not knowing the definition of "0" on the modular curve in this context, which I was taking to be the preimage of the identity on the Jacobian.]

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Jonah Sinick
Jonah Sinick
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I only know this material in a vague and impressionistic way, but I think that it the fact that $0$ goes to the identity on the elliptic curve follows from Eichler-Shimura theory.

If $E/\mathbb{Q}$ is parametrized by a modular curve, then it's a quotient of the Jacobian of the modular curve. In the composite map $$X_{0}(N) \to J_{0}(N) \to E$$ each arrow sends $0$ to $0$.

[Edit: Looking at this question I see that apparently 0 need not go to the identity in general. So now I'm puzzled. I think that my confusion might come from not knowing the definition of "0" on the modular curve in this context, which I was taking to be the preimage of the identity on the Jacobian.]

I only know this material in a vague and impressionistic way, but I think that it the fact that $0$ goes to the identity on the elliptic curve follows from Eichler-Shimura theory.

If $E/\mathbb{Q}$ is parametrized by a modular curve, then it's a quotient of the Jacobian of the modular curve. In the composite map $$X_{0}(N) \to J_{0}(N) \to E$$ each arrow sends $0$ to $0$.

I only know this material in a vague and impressionistic way, but I think that it the fact that $0$ goes to the identity on the elliptic curve follows from Eichler-Shimura theory.

If $E/\mathbb{Q}$ is parametrized by a modular curve, then it's a quotient of the Jacobian of the modular curve. In the composite map $$X_{0}(N) \to J_{0}(N) \to E$$ each arrow sends $0$ to $0$.

[Edit: Looking at this question I see that apparently 0 need not go to the identity in general. So now I'm puzzled. I think that my confusion might come from not knowing the definition of "0" on the modular curve in this context, which I was taking to be the preimage of the identity on the Jacobian.]

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Jonah Sinick
Jonah Sinick
  • 7.1k
  • 6
  • 43
  • 77

I only know this material in a vague and impressionistic way, but I think that it the fact that $0$ goes to the identity on the elliptic curve follows from Eichler-Shimura theory.

If $E/\mathbb{Q}$ is parametrized by a modular curve, then it's a quotient of the Jacobian of the modular curve. In the composite map $$X_{0}(N) \to J_{0}(N) \to E$$ each arrow sends $0$ to $0$.