Local Zeta Functions From Calabi-Yau Differential Equations

Understand

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.