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Abstract

Let $E$ be an elliptic curve defined over the rational numbers. We consider the infinite extension $\mathbb Q (E_{\mathrm {tor}})$ of the rational numbers obtained by adjoining to $\mathbb Q $ all $x$- and $y$-coordinates of torsion points of $E$ with respect to a fixed Weierstrass model over $\mathbb Q $. We prove that the height of an element in $\mathbb Q (E_{\mathrm {tor}})$ is either zero or bounded below by an explicit absolute constant that only depends on the conductor of $E$. An important intermediate step is to find an explicit upper bound for a supersingular prime for $E$ (an explicit version of Elkies’ Theorem).