Automatic Sequences and Curves over Finite Fields

We prove that if $y={\sum }_{n=0}^{\mathrm{\infty }}\mathbf{a}(n)x^n\in {\mathbb{F}}_q[[x]]$ is an algebraic power series of degree $d$, height $h$, and genus $g$, then the sequence $\mathbf{a}$ is generated by an automaton with at most $q^{h+d+g-1}$ states, up to a vanishingly small error term. This is a significant improvement on previously known bounds. Our approach follows an idea of David Speyer to connect automata theory with algebraic geometry by representing the transitions in an automaton as twisted Cartier operators on the differentials of a curve.