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

Abstract

A k-arc in the projective space $$\\mathrm{PG}(n,q)$$PG(n,q) is a set of k projective points such that no subcollection of $$n+1$$n+1 points is contained in a hyperplane. In this paper, we construct new 60-arcs and 110-arcs in $$\\mathrm{PG}(4,q)$$PG(4,q) that do not arise from rational or elliptic curves. We introduce computational methods that, when given a set $$\\mathcal {P}$$P of projective points in the projective space of dimension n over an algebraic number field $${\\mathbb {Q}}(\\xi )$$Q(ξ), determines a complete list of primes p for which the reduction modulo p of $$\\mathcal {P}$$P to the projective space $$\\mathrm{PG}(n,p^h)$$PG(n,ph) may fail to be a k-arc. Using these methods, we prove that there are infinitely many primes p such that $$\\mathrm{PG}(4,p)$$PG(4,p) contains a $$\\mathrm{PSL}(2,11)$$PSL(2,11)-invariant 110-arc, where $$\\mathrm{PSL}(2,11)$$PSL(2,11) is given in one of its natural irreducible representations as a subgroup of $$\\mathrm{PGL}(5,p)$$PGL(5,p). Similarly, we show that there exist $$\\mathrm{PSL}(2,11)$$PSL(2,11)-invariant 110-arcs in $$\\mathrm{PG}(4,p^2)$$PG(4,p2) and $$\\mathrm{PSL}(2,11)$$PSL(2,11)-invariant 60-arcs in $$\\mathrm{PG}(4,p)$$PG(4,p) for infinitely many primes p.