Variation of the Faltings height in a \(\overline{\mathbb{Q}}\)-isogeny class of elliptic curves
This paper proves some bounds on the variation of the Faltings height of a non-CM elliptic curve within a \(\overline{\mathbb Q}\)-isogeny class. The principal result in this direction is the following. Let \(E\) be a semi-stable elliptic curve over a number field \(K\) without complex multiplication. Let \(n\) be a positive integer, and let \(\prod_{i=1}^r p_i^{\alpha_i}\) be its prime factorization. Let \(P\) be a torsion point of \(E\) of order exactly \(n\), and let \(\pi\:E\to E'\) be the \(K(P)\)-isogeny whose kernel is generated by \(P\). Then \[ h(E')=h(E)+\tfrac 12\log n-\sum_i ((p_i^{\alpha_i}-1)/((p_i^2-1)p_i^{\alpha_i-1}))\log p_i + O(1), \] where \(O(1)\) depends only on \(E\). Moreover, if \(\text{Gal}(\overline K/K)\) acts transitively on the set of points of order \(n\) of \(E\), then the \(O(1)\) can be omitted. The paper also proves a similar result for the norm of the conductor.
scientific article; zbMATH DE number 1425197
scientific article