STNB2020(34th edition)

Kummer Theory for Elliptic Curves


Sebastiano Tronto


Consider an elliptic curve $E$ over a number field $K$, and a point $a$ in $E(K)$ of infinite order. We are interested in the field extensions of $K$ that are generated by the coordinates of the $N$-division points of $a$, for any integer $N$. In particular, we would like to compute their degree over the $N$-torsion field of $E$. It is known that the ratio between $N^2$ and this degree is bounded independently of $N$. In a joint work with D. Lombardo we give an explicit value for this bound, in case ${\rm End}_K(E)$ is trivial, in terms of the $l$-adic Galois representations associated with $E/K$ and of simple divisibility properties of the point $a$.


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