Vector-Valued Period Polynomials and Zeta Values of Quadratic Fields

TL;DR

This work constructs and analyzes vector-valued period polynomials attached to level $N$ cusp forms built from $\Gamma_{0}(N)$-orbits of binary quadratic forms. It provides a closed formula for these period polynomials, separating an algebraic part from a zeta-part built from $\zeta_{N,D,\rho}(s)$, and uses Fricke symmetry to derive explicit relations among zeta-values at critical integers. The authors then give a precise odd-$k$ difference formula for $\zeta_{N,D,\rho}(k)-\zeta_{N,D,-\rho}(k)$ in terms of Bernoulli numbers and quadratic-form sums, and, under a vanishing condition on Fricke-invariant cusp forms, obtain a finite divisor-sum expression for $\zeta_{\mathbb{Q}(\sqrt{D})}(k)$ at even $k$. Collectively, these results illustrate a powerful period-zeta connection at level $N$ that yields computable formulas for Dedekind zeta values via quadratic-form data and ideal-class analyses.

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 $ρ$ with $ρ^2\equiv D\pmod{4N}$. In this paper, we consider two weight $2k$ cusp forms $f^{\pm}_{k,N,D,ρ}$ on $Γ_0(N)$ 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 \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 $ζ_{\mathbb{Q}(\sqrt{D})}(k)$ at even integers $k$.

Theorems & Definitions (25)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Example 2.1
• Proposition 3.1
• proof
• Lemma 4.1
• proof
• proof : Proof of Theorem \ref{['thm-period']}
• Lemma 5.1