Fibre products of supersingular curves and the enumeration of irreducible polynomials with prescribed coefficients

For any positive integers n ź 3 , r ź 1 we present formulae for the number of irreducible polynomials of degree n over the finite field F 2 r where the coefficients of x n - 1 , x n - 2 and x n - 3 are zero. Our proofs involve counting the number of points on certain algebraic curves over finite fields, a technique which arose from Fourier-analysing the known formulae for the F 2 base field cases, reverse-engineering an economical new proof and then extending it. This approach gives rise to fibre products of supersingular curves and makes explicit why the formulae have period 24 in n.