Traces of Hecke operators and refined weight enumerators of Reed-Solomon codes

We study the quadratic residue weight enumerators of the dual projective Reed-Solomon codes of dimensions 5 5 and q − 4 q-4 over the finite field F q \mathbb {F}_q . Our main results are formulas for the coefficients of the quadratic residue weight enumerators for such codes. If q = p v q=p^v and we fix v v and vary p p , then our formulas for the coefficients of the dimension q − 4 q-4 code involve only polynomials in p p and the trace of the q t h q^{\mathrm {th}} and ( q / p 2 ) th (q/p^2)^{\text {th}} Hecke operators acting on spaces of cusp forms for the congruence groups SL 2 ⁡ ( Z ) , Γ 0 ( 2 ) \operatorname {SL}_2 (\mathbb {Z}), \Gamma _0(2) , and Γ 0 ( 4 ) \Gamma _0(4) . The main tool we use is the Eichler-Selberg trace formula, which gives along the way a variation of a theorem of Birch on the distribution of rational point counts for elliptic curves with prescribed 2 2 -torsion over a fixed finite field.