Bloch’s formula for 0-cycles with modulus and higher-dimensional class field theory

Abstract

We prove Bloch’s formula for the Chow group of 0-cycles with modulus on a smooth quasi-projective surface over a field. We use this formula to give a simple proof of the rank one case of a conjecture of Deligne and Drinfeld on lisse Q ¯ ℓ \overline {\mathbb {Q}}_{\ell } -sheaves. This was originally solved by Kerz and Saito in characteristic ≠ 2 \neq 2 .