Characteristic Epsilon Cycles of $\mathrm{\ell }$-adic Sheaves on Varieties

Let $X$ be a smooth variety over a finite field ${\mathbb{F}}_q$. Let $\mathrm{\ell }$ be a rational prime number invertible in ${\mathbb{F}}_q$. For an $\mathrm{\ell }$-adic sheaf $\mathcal{F}$ on $X$, we construct a cycle supported on the singular support of $\mathcal{F}$ whose coefficients are $\mathrm{\ell }$-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.