Periodic Continued Fractions and Hyperelliptic Curves

We investigate when an algebraic function of the form φ(λ) = −B(λ)+ √ R(λ) A(λ) , where R(λ) is a polynomial of odd degree N = 2g + 1 with coefficients in C, can be written as a periodic α-fraction of the form φ(λ) = [b0; b1, b2, . . . , bN ]α = b0 + λ− α1 b1 + λ−α2 b2+ . . .bN−1+ λ−αN bN+ λ−α1 b1+ λ−α2 b2+ . . . , for some fixed sequence αi. We show that this problem has a natural answer given by the classical theory of hyperelliptic curves and their Jacobi varieties. We also consider pure periodic α-fraction expansions corresponding to the special case when bN = b0.