Abstract: The singular support and the characteristic cycle of a constructible sheaf on a smooth variety were defined on the cotangent bundle by Beilinson and Saito, respectively. To consider the characteristic cycle in the mixed characteristic case, as a replacement of the cotangent bundle, Saito defined the Frobenius-Witt cotangent bundle of a regular flat scheme over a discrete valuation ring of mixed characteristic as a vector bundle on the closed fiber. This talk consists of two parts. In the first part, we review the definitions of the FW-cotangent bundle and the singular support. In the second part, we define the F-characteristic cycle satisfying a conductor formula of a rank one sheaf on an arithmetic surface on the FW-cotangent bundle. The definition is based on the computation of the characteristic cycle in the geometric case by Yatagawa. We discuss some properties of the characteristic form, which are necessary for the definition of the F-characteristic cycle.
