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

For any positive integers n≥3n≥3, r≥1r≥1 we present formulae for the number of irreducible polynomials of degree n   over the finite field F2rF2r where the coefficients of xn−1xn−1, xn−2xn−2 and xn−3xn−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 F2F2 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.