Sharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields

In this paper we study extension theorems associated with general varieties in two dimensional vector spaces over finite fields. Applying Bezouts theorem, we obtain the sufficient and necessary conditions on general curves where sharp $L^p-L^r$ extension estimates hold. Our main result can be considered as a nice generalization of works by Mochenhaupt and Tao and Iosevich and Koh. As an application of our sharp extension estimates, we also study the Falconer distance problems in two dimensions.