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Explicit small heights in infinite non-abelian extensions

Volume 199 / 2021

Linda Frey Acta Arithmetica 199 (2021), 111-133 MSC: Primary 11G05; Secondary 11G50. DOI: 10.4064/aa190514-15-1 Published online: 28 May 2021


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).


  • Linda FreyDepartment of Mathematical Sciences
    University of Copenhagen
    Universitetsparken 5
    2100 København, Denmark

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