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 varietiesin two dimensional vector spaces over finite fields. Applying Bezout's 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 nicegeneralization of works by Mochenhaupt and Tao and Iosevich and Koh. As anapplication of our sharp extension estimates, we also study the Falconerdistance problems in two dimensions.