Vector-Valued Period Polynomials and Zeta Values of Quadratic Fields
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arXiv
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| Format: | Preprint |
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2026
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| author | Kwon, Yeong-Wook Lim, Subong Raji, Wissam |
| author_facet | Kwon, Yeong-Wook Lim, Subong Raji, Wissam |
| contents | 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 $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_00572 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vector-Valued Period Polynomials and Zeta Values of Quadratic Fields Kwon, Yeong-Wook Lim, Subong Raji, Wissam Number Theory 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 $k$. |
| title | Vector-Valued Period Polynomials and Zeta Values of Quadratic Fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2602.00572 |