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In the Proof of Mordell Conjecture by Gerd Faltings, it is famous that Parshin constructed a curve $C_P$ for each $\mathbb{Q}$-rational point $P$ on the given curve $C$ over $\mathbb{Q}$ such that the genus of $C$ satisfies $g(C) > 0$.

Is there anybody that, in case where $C = E$ is an elliptic curve over $Q$, gives me an explicit construction

$C_P \to E$

together with a $\mathbb{Q}$-rational point $P$ on $E$?

I am quite anxious for an explicit example of Parshin's construction. Many thanks. Sincerely yours, Pierre MATSUMI

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  • $\begingroup$ Would your question be answered by Lemma 5.3 in J. S. Milne's text on abelian varieties, and the discussion preceding it? Here is the link: jmilne.org/math/CourseNotes/AV.pdf $\endgroup$
    – Vesselin Dimitrov
    Commented Sep 27, 2013 at 15:37

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If I am not mistaken, you can find such an example in the following lecture notes by Akshay:

http://virtualmath1.stanford.edu/~conrad/mordellsem/Notes/L01.pdf

(to be precise, see the bottom part of page 4).

I realized this answer is late by ~10 years. So not sure if this is still relevant to OP.

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