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

TL;DR

This paper constructs new large arcs in projective 4-space invariant under PSL(2,11), introduces computational methods to analyze their properties over various primes, and proves their existence for infinitely many primes.

Contribution

It introduces novel computational techniques to identify PSL(2,11)-invariant k-arcs in PG(4,q) and proves their existence for infinitely many primes, expanding the understanding of arc constructions.

Findings

Constructed 60- and 110-arcs in PG(4,q) not from rational or elliptic curves.

Developed methods to determine primes where reductions preserve arc properties.

Proved the existence of infinitely many primes with PSL(2,11)-invariant arcs in PG(4,p) and PG(4,p^2).

Abstract

A \textit{k}-arc in the projective space ${\rm PG}(n,q)$ 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 ${\rm PG}(4,q)$ 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 $\mathcal{Q}(\xi)$, determines a complete list of primes $p$ for which the reduction modulo $p$ of $\mathcal{P}$ to the projective space ${\rm PG}(n,p^h)$ may fail to be a $k$-arc. Using these methods, we prove that there are infinitely many primes $p$ such that ${\rm PG}(4,p)$ contains a ${\rm PSL}(2,11)$-invariant $110$-arc, where ${\rm PSL}(2,11)$ is given in one of its natural irreducible representations as a subgroup of ${\rm PGL}(5,p)$. Similarly, we show that there exist ${\rm PSL}(2,11)$-invariant $110$-arcs in ${\rm PG}(4,p^2)$ and ${\rm PSL}(2,11)$-invariant $60$-arcs in ${\rm PG}(4,p)$ for infinitely many primes $p$.