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

  • Torger Olson
  • Eric Swartz


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.


Finite projective space k-arc \(\mathrm{PSL}(2, 11)\) 

Mathematics Subject Classification

51E20 20G40 



The authors would like to thank John Bamberg for helpful discussions about the \({\mathrm{GAP}}\) package FinInG and Jan De Beule for \({\mathrm{GAP}}\) code that allows for calculations using FinInG beyond what is normally possible in \({\mathrm{GAP}}\). The authors would also like to thank the anonymous referees for many useful suggestions that improved the readability of this paper.


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Authors and Affiliations

  1. 1.Department of MathematicsCollege of William & MaryWilliamsburgUSA

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