Zeta functions for families of Calabi-Yau \(n\)-folds with singularities (Q2918490)

Relations beween the occurring singularity structure and the decomposition of the local zeta function in families of Calabi-Yau \(n\)-folds containing singular fibres are studied. Here the local zeta function for a smooth projective variety \(X\) over \(\mathbb{F}_p\) is defined s follows: NEWLINE\[NEWLINE \varsigma(X|\mathbb{F}_p, t):=\exp(\sum\limits_{r\in \mathbb{N}}\# X(\mathbb{F}_{p^r})\frac{t^r}{r}). NEWLINE\]NEWLINE Properties about the local zeta functions at good primes are listed. The singularities occurring in some \(1\)--dimensional and \(2\)--dimensional families (families of Fermat-type Calab-Yau \(n\)-folds) are analysed in detail. Examples of \(2\)--parameter families are especially studied. Here the singularity analysis provides correct predictions for the changes of degree in the decomposition of the zeta--function when passing to singular fibres.NEWLINENEWLINEFor the entire collection see [Zbl 1242.11004].