Let $X$ be a smooth variety over a finite field $\mathbb{ F}_q$. Let $\ell$ be a rational prime number invertible in $\mathbb{ F}_q$. For an $\ell$-adic sheaf $\mathcal{ F}$ on $X$, we construct a cycle supported on the singular support of $\mathcal{ F}$ whose coefficients are $\ell$-adic numbers modulo roots of unity. It is a refinement of the characteristic cycle $CC(\mathcal{ F})$, in the sense that it satisfies a Milnor-type formula for local epsilon factors. After establishing fundamental results on the cycles, we prove a product formula of global epsilon factors modulo roots of unity. We also give a generalization of the results to varieties over general perfect fields.
Categories: math.AG math.NT