Vector-Valued Period Polynomials and Zeta Values of Quadratic Fields

Technical Summary: Vector-Valued Period Polynomials and Zeta Values of Quadratic Fields

Problem Statement
The paper addresses the arithmetic of special values of zeta functions and $L$-functions, specifically focusing on the Dedekind zeta function $\zeta_{\mathbb{Q}(\sqrt{D})}(s)$ of real quadratic fields. While the connection between modular forms and zeta values is well-established (notably through Hecke's foundational work), the authors aim to make these connections explicit in the presence of level structure ($\Gamma_0(N)$). The central problem is to derive closed-form expressions for zeta values at integer arguments by utilizing the theory of vector-valued period polynomials associated with cusp forms constructed from binary quadratic forms. Specifically, the paper seeks to separate the algebraic and transcendental components of these values and to evaluate differences between zeta values corresponding to different choices of square roots of the discriminant modulo $4N$.

Methodology
The authors employ a synthesis of the theory of modular forms, binary quadratic forms, and period polynomials. The methodology proceeds through the following steps:

1. Construction of Cusp Forms: For integers $k \ge 2$, $N \ge 1$, and a positive integer $D$ congruent to a square modulo $4N$ (with a fixed square root $\rho$), the authors define explicit weight $2k$ cusp forms $f^{\pm}_{k,N,D,\rho}$ on $\Gamma_0(N)$. These forms are defined as weighted sums over the set of binary quadratic forms $Q_{N,D,\rho}$, which consists of forms $[Na, b, c]$ with discriminant $D$ and specific congruence conditions on the middle coefficient ($b \equiv \rho \pmod{2N}$).
2. Vector-Valued Period Polynomials: The authors investigate the vector-valued period polynomial $r_f$ associated with these cusp forms. Unlike scalar period polynomials, the vector-valued version tracks cosets in $\Gamma_0(N) \backslash \text{SL}_2(\mathbb{Z})$, allowing for the separation of arithmetic contributions that are invisible in the scalar setting. The analysis involves decomposing these polynomials into even and odd parts ($r_f^+$ and $r_f^-$).
3. Integral Representations and Limits: The core technical machinery involves evaluating integrals of the form $\int (f|_{2k}A)(z)(X-z)^{2k-2} dz$. By expressing the cusp forms as sums over quadratic forms, the authors convert these integrals into limits of sums involving powers of quadratic forms. They utilize contour integration and residue calculus to isolate contributions from specific regions of the quadratic form parameters (e.g., $a>0>c$ vs. $a<0<c$).
4. Symmetry and Functional Equations: The paper leverages the Fricke involution $W_N$ and the associated functional equations of the $L$-functions. By analyzing the behavior of the period polynomials under these involutions, particularly for odd and even weights, the authors derive relations between the coefficients of the period polynomials and the zeta values.

Key Contributions and Results

• Closed Formula for Vector-Valued Period Polynomials (Theorem 2.1): The primary result is an explicit closed formula for the vector-valued period polynomial $r^+_{f^+} + r^-_{f^-}$. The identity component of this formula is particularly significant as it separates into two distinct parts:

  • An algebraic part arising from finite sums over binary quadratic forms where the coefficients satisfy specific sign conditions ($a>0>c$ and $a<0<c$).
  • A zeta part involving the values at $s=k$ of partial zeta functions $\zeta_{N,D,\rho}(s)$ constructed from $\Gamma_0(N)$-equivalence classes.
    This establishes a "level-$N$ period–zeta connection," expressing period polynomial coefficients in terms of quadratic-form sums and explicitly normalized zeta values.

• Evaluation of Zeta Value Differences (Theorem 2.2): For odd integers $k$, the authors derive a closed formula for the difference $\zeta_{N,D,\rho}(k) - \zeta_{N,D,-\rho}(k)$. This result isolates the contribution of the congruence condition $\rho$ and demonstrates how Fricke symmetry and the even/odd decomposition of periods translate into a non-trivial arithmetic identity. The difference is expressed explicitly in terms of Bernoulli numbers, a finite sum over quadratic forms, and powers of $N$ and $D$.

• Finite Divisor-Sum Formula for Dedekind Zeta Values (Theorem 2.3): Under a vanishing condition on Fricke-invariant cusp forms at lower levels (specifically, assuming $S^+_{2k}(N/d) = \{0\}$ for all divisors $d$ of $N$), the paper provides an explicit finite divisor-sum formula for the Dedekind zeta value $\zeta_{\mathbb{Q}(\sqrt{D})}(k)$ at even integers $k$. This formula refines classical Hecke-type evaluations by incorporating congruence restrictions and level structure, expressing the zeta value as a sum involving divisor functions $\sigma_{k-1}$ and congruence conditions on the middle coefficient of the quadratic forms.

Significance and Claims
The paper claims to make the connections between modular forms, binary quadratic forms, and algebraic invariants of number fields explicit. By computing vector-valued period polynomials in closed form, the authors provide a mechanism to produce explicit expressions for zeta values that reveal their symmetries.

The authors position their work as a refinement of existing theories:

• They extend the classical theory of period polynomials to the vector-valued setting to handle level structures effectively.
• They demonstrate how sums over quadratic forms of fixed discriminant can be used to obtain explicit formulas for arithmetic quantities, building on the work of Zagier.
• The derived formulas for $\zeta_{\mathbb{Q}(\sqrt{D})}(k)$ are presented as computable expressions that illustrate the strength of period methods when level structure is present.

The paper concludes by providing concrete examples (evaluating $\zeta_{\mathbb{Q}(\sqrt{17})}(2)$ and $\zeta_{\mathbb{Q}(\sqrt{145})}(2)$) to demonstrate the applicability of the derived theorems, confirming that the vanishing conditions on lower-level cusp forms allow for the explicit calculation of these values using the proposed finite divisor-sum formulas.