Vector-valued period polynomials and zeta values of quadratic fields

Let k≥ 2 k ≥ 2 and N≥ 1 N ≥ 1 be integers. Let D be a positive integer that is congruent to a square modulo 4 N , and fix ρ ρ with ρ ²≡ D 4N ρ 2 ≡ D ( mod 4 N ) . In this paper, we consider two weight 2 k cusp forms f±k,N,D,ρ f k , N , D , ρ ± on Γ ₀(N) Γ 0 ( N ) defined by sums over binary quadratic forms and investigate the vector-valued period polynomial arising from these forms. Our first main result gives a closed formula for this vector-valued period polynomial. The identity component of this formula is particularly explicit: It separates as the sum of a finite algebraic part coming from some binary forms and a zeta part involving the values at $$s=k$$ s = k of certain zeta functions. Using this formula together with a symmetry of vector-valued period polynomials, we explicitly evaluate, for odd k , the difference between the zeta values corresponding to the two choices of square root of D modulo 4 N , in terms of Bernoulli numbers and a finite quadratic form sum. Finally, under a vanishing condition on Fricke-invariant cusp forms at lower levels, we obtain a finite divisor sum formula for the Dedekind zeta values ζ Q(√D)(k) ζ Q ( D ) ( k ) at even integers k .