Local Zeta Functions From Calabi-Yau Differential Equations

TL;DR

This paper presents a fast computational method to determine local zeta functions of Calabi-Yau manifolds using their Picard-Fuchs equations, with results for several examples and primes, revealing links to modular forms and singularities.

Contribution

It introduces a practical approach to compute zeta functions of Calabi-Yau families directly from differential operators, including cases with singularities and modular form connections.

Findings

Computed zeta-function numerators for six Calabi-Yau manifolds across multiple primes.

Identified factorization patterns linked to singularities and modular forms.

Revealed attractor points where the Hodge structure splits.

Abstract

The zeta-function of a manifold is closely related to, and sometimes can be calculated completely, in terms of its periods. We report here on a practical and computationally rapid implementation of this procedure for families of Calabi-Yau manifolds with one complex structure parameter phi. Although partly conjectural, it turns out to be possible to compute the matrix of the Frobenius map on the third cohomology group of X(phi) directly from the Picard-Fuchs differential operator of the family. To illustrate our method, we compute tables of the quartic numerators of the zeta-functions for six manifolds of increasing complexity as the parameter phi varies in Fp. For four of these manifolds, we do this for the 500 primes p=5,7,...,3583, while for two manifolds we extend the calculation to 1000 primes. The tables for 5 <= p <= 97 are part of this article while the remaining tables are attached in electronic form. Interest attaches to the cases for which the numerators factorise. Some of these factorisations can be associated with parameter values for which the underlying manifold becomes singular. For the cases we consider here, the singularities are all of conifold or hyperconifold type. In these cases the numerator degenerates to a cubic and this factorises into the product of a linear and a quadratic factor. The quadratic term contains a coefficient that is the p'th coefficient of a modular form. Some of our examples have singularities when the parameter satisfies a polynomial equation that does not factorise over Q. When this happens, the corresponding forms are modular forms with neben type or Hilbert modular forms. The numerator can also factorise into two quadrics. This happens when the Hodge structure of the manifold splits, sometimes this happens for algebraic values of the parameter and we identify, in this way, attractor points of rank two of the parameter space.