Description
Online Number Theory Seminar
Abstract: Let $p$ be a prime. In this talk we review the $\mathbb{Z}_p$ cyclotomic extension of $\mathbb{Q}$ (often denoted by $\mathbb{Q}_{\infty}$), and what is known or conjectured regarding Diophantine equations over $\mathbb{Q}_{\infty}$. In particular, and contrary to expectations, we show that the unit equation can have infinitely many solutions over $\mathbb{Q}_{\infty}$. We use this fact to construct infinite families of hyperelliptic curves, defined over $\mathbb{Q}_{\infty}$, with good reduction away from 2 and $p$, thereby showing that the Shafarevich conjecture does not generalise to this setting. This talk is based on joint work with Robin Visser.
For access please contact the organizers (ntrg[at]science.unideb.hu).