Primitive divisors of sequences associated to elliptic curves over function fields

Abstract: In the first part of the talk, we give a gentle introduction into the subject of divisibility sequences over the rational numbers and discuss the notion of a primitive divisor/Zsigmondy bound. We then explain how these notions can be extended to number fields and function fields, and how to obtain a divisibility sequence from a non-torsion point on an elliptic curve over any of these fields. There will also be plenty of nice examples.

In the second part of the talk, we discuss the typical methods that are used to prove the existence of a Zsigmondy bound for a divisibility sequence obtained from a non-torsion point on an elliptic curve $E$ over a number or function field $K$. Let $P$ be this non-torsion point in $E(K)$, and suppose Q is a torsion point in $E(K)$. We can also associate a sequence of divisors $\{D_{nP+Q}\}$ on $K$ to the sequence of points $\{nP+Q\}$. In my preprint, we proved the existence of a Zsigmondy bound for this sequence $\{D_{nP+Q}\}$ for $K$ a function field (under some minor conditions), extending the analogous result of Verzobio over number fields. I will provide the crucial ideas to apply the existing methods of the case $\{nP\}$ to my case $\{nP+Q\}$. Additionally, I will highlight the differences with the number field case.

number theory

Audience: researchers in the topic

Warsaw Number Theory Seminar