Transitive PSL(2,11)-invariant k-arcs in PG(4,q)

Abstract

A k-arc in the projective space $\mathrm{PG}(n,q)$ is a set of k projective points such that no subcollection of $n+1$ points is contained in a hyperplane. In this paper, we construct new 60-arcs and 110-arcs in $\mathrm{PG}(4,q)$ that do not arise from rational or elliptic curves. We introduce computational methods that, when given a set $\mathcal {P}$ of projective points in the projective space of dimension n over an algebraic number field ${\mathbb {Q}}(\xi )$, determines a complete list of primes p for which the reduction modulo p of $\mathcal {P}$ to the projective space $\mathrm{PG}(n,p^h)$ may fail to be a k-arc. Using these methods, we prove that there are infinitely many primes p such that $\mathrm{PG}(4,p)$ contains a $\mathrm{PSL}(2,11)$-invariant 110-arc, where $\mathrm{PSL}(2,11)$ is given in one of its natural irreducible representations as a subgroup of $\mathrm{PGL}(5,p)$. Similarly, we show that there exist $\mathrm{PSL}(2,11)$-invariant 110-arcs in $\mathrm{PG}(4,p^2)$ and $\mathrm{PSL}(2,11)$-invariant 60-arcs in $\mathrm{PG}(4,p)$ for infinitely many primes p.