  We prove a duality theorem for the $p$-adic etale motivic cohomology of a
variety $U$ which is the complement of a divisor on a smooth projective variety
over $\F_p$. This extends the duality theorems of Milne and Jannsen-Saito-Zhao.
The duality introduces a filtration on $H^1_{\etl}(U, {\Q}/{\Z})$. We identify
this filtration to the classically known Matsuda filtration when the reduced
part of the divisor is smooth. We prove a reciprocity theorem for the idele
class groups with modulus introduced by Kerz-Zhao and Rulling-Saito. As an
application, we derive the failure of Nisnevich descent for Chow groups with
modulus.
