Local Zeta Functions of Multiparameter Calabi-Yau Threefolds from the Picard-Fuchs Equations

TL;DR

This work extends the deformation-based computation of local zeta functions from one to multiple complex-structure parameters in Calabi–Yau threefolds. By developing an enhanced Picard–Fuchs framework and a structured basis of sections, the authors derive an efficient p-adic Frobenius action matrix $U_p(m u)$ that yields the zeta numerator via $R_p(X_{m u},T)= ext{det}(I - T U_p(m u))$, while carefully handling conifold and apparent singularities and coordinate changes. The approach is demonstrated on concrete multiparameter families (two-parameter mirror octics, a two-parameter split quintic, and the five-parameter Hulek–Verrill manifolds), with results agreeing with prior computations and extending them to higher primes; a Mathematica package CY3Zeta is provided to implement the methods. The paper also details the role of mirror symmetry, the Gauss–Manin connection, and $p$-adic techniques (including Teichmüller lifts and the $p$-adic $ ext{G}$amma-class) in structuring and accelerating the computations, and outlines the limitations and potential extensions to higher dimensions or more singular moduli points.

Abstract

The deformation approach of arXiv:2104.07816 for computing zeta functions of one-parameter Calabi-Yau threefolds is generalised to cover also multiparameter manifolds. Consideration of the multiparameter case requires the development of an improved formalism. This allows us, among other things, to make progress on some issues left open in previous work, such as the treatment of apparent and conifold singularities and changes of coordinates. We also discuss the efficient numerical computation of the zeta functions. As examples, we compute the zeta functions of the two-parameter mirror octic, a non-symmetric split of the quintic threefold also with two parameters, and the $S_5$ symmetric five-parameter Hulek-Verrill manifolds. These examples allow us to exhibit the several new types of geometries for which our methods make practical computations possible. They also act as consistency checks, as our results reproduce and extend those of arXiv:hep-th/0409202 and arXiv:math/0304169. To make the methods developed here more approachable, a Mathematica package "CY3Zeta" for computing the zeta functions of Calabi-Yau threefolds, which is attached to this paper, is presented.

Figures (2)

• Figure 1: The vector bundle $\mathcal{H}$.
• Figure 2: A heuristic sketch of the complex structure moduli space $\mathcal{M}_{\mathbbl{C} S}$. Each point $\bm \varphi$ in the moduli space correspond to a Calabi--Yau manifold $X_{\bm \varphi}$. The fibre above each point is the middle cohomology group $H^3(X_{\bm \varphi})$ of the corresponding manifold. The dashed circle represents the $p$-adic unit disk $||x||_p < 1$, with $1$ also being the radius of convergence of the power series appearing in the expansions of the periods and their derivatives (the logarithms appearing in the periods drop out of the expression for the matrix $\text{U}_p(\bm \varphi)$). At a generic point $\varphi$, the Frobenius map $\text{Fr}_p$ acts between two distinct fibres $H^3(X_{\bm \varphi})$ and $H^3(X_{\bm \varphi^p})$, which in particular implies that the matrix $\text{U}_p(\bm \varphi)$ is not well-defined. At Teichmüller representatives $\text{Teich}(\bm n)$ of integral points $\bm n \in \mathbbl{Z}^m$ the fibres coincide, and the matrix $\text{U}_p(\bm \varphi)$ is well-defined. The Teichmüller representatives lie at the boundary of the $p$-adic unit disk, where the period series do not converge, but the matrix $\text{U}_p(\bm \varphi)$ does.