Vector-Valued Period Polynomials and Zeta Values of Quadratic Fields

Abstract

Let $k\ge 2$ and $N\ge 1$ be integers. Let $D$ be a positive integer that is congruent to a square modulo $4N$, and fix $\rho$ with ${\rho }^2\equiv D(\mathrm{mod}4N)$. In this paper, we consider two weight $2k$ cusp forms $f_{k,N,D,\rho }^{\pm }$ on ${\Gamma }_0(N)$ 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 \textit{algebraic part} coming from some binary forms and a \textit{zeta part} involving the values at $s=k$ 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 $4N$, 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 ${\zeta }_{\mathbb{Q}(\sqrt{D})}(k)$ at even integers $k$.