del Pezzo surfaces with one bad prime over cyclotomic ${\mathbb{Z}}_{\ell }$-extensions

Abstract

Let $K$ be a number field and $S$ a finite set of primes of $K$. Scholl proved that there are only finitely many $K$-isomorphism classes of del Pezzo surfaces of any degree $1\le d\le 9$ over $K$ with good reduction away from $S$. Let instead $K$ be the cyclotomic ${\mathbb{Z}}_5$-extension of $\mathbb{Q}$.In this paper, we show, for $d=3$, $4$, that there are infinitely many $\overline{\mathbb{Q}}$ isomorphism classes of del Pezzo surfaces, defined over $K$, with good reduction away from the unique prime above $5$.