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In the famous proof of the Mordell conjecture by Gerd Faltings, the so-called Parshin construction is known.

For example, let $E/\mathbb{Q}$ be an smooth elliptic curve, and let us pick up a $\mathbb{Q}$-rational point $P$ on $E$ (i.e. $x, y$ coordinates are both lying in $\mathbb{Q}$).

According to Parshin's construction, we can associate the covering $p:C \to E$ to $P$.

Could you please provide me with any "explicit" example of this covering $p$?

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    $\begingroup$ You should have a look at the chapter on the Parshin construction in the Astérisque volume on the Mordell conjecture (edited by L. Szpiro). $\endgroup$
    – Damian Rössler
    Commented Aug 5, 2013 at 13:28

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