$\ell$-independence of the trace of local monodromy in a relative case

TL;DR

This paper proves that for a family of varieties, the sum of traces of local monodromy on cohomology is an integer independent of the prime , extending known results to a relative setting.

Contribution

It establishes the -independence of the trace sum of local monodromy in a relative family of varieties, generalizing previous results.

Findings

The sum of traces is an integer.

The trace sum is independent of .

Applicable to families of varieties.

Abstract

For a family of varieties, we prove that the alternating sum of the traces of "local" monodromy acting on the $\ell$-adic \'etale cohomology groups of the generic fiber is an integer which is independent of $\ell$.