A modular equality for Cameron–Liebler line classes

In this paper we prove that a Cameron–Liebler line class L in PG ( 3 , q ) with parameter x has the property that ( x 2 ) + n ( n − x ) ≡ 0 mod q + 1 for the number n of lines of L in any plane of PG ( 3 , q ) . It follows that the modular equation ( x 2 ) + n ( n − x ) ≡ 0 mod q + 1 has an integer solution in n. This result rules out roughly at least one half of all possible parameters x. As an application of our method, we determine the spectrum of parameters of Cameron–Liebler line classes of PG ( 3 , 5 ) . This includes the construction of a Cameron–Liebler line class with parameter 10 in PG ( 3 , 5 ) and a proof that it is unique up to projectivities and dualities.