A theorem of Christol states that a power series $y$ over a finite field is an algebraic function if and only if its coefficient sequence can be produced by a finite automaton, which is a limited model of a computer with no memory. The proof uses combinatorics and linear algebra, but hidden in the theorem there is geometric information about a curve that contains $y$ in its function field. I make this explicit by demonstrating a precise link between the complexity of the automaton and the geometry of the curve.