## STNB2020(34th edition)

### Kummer Theory for Elliptic Curves

#### Presenters

Sebastiano Tronto

#### Abstract

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 $a$.

#### Files

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