Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function

M. Toda in 1967 (\textit{J. Phys. Soc. Japan}, \textbf{22} and \textbf{23}) considered a lattice model with exponential interaction and proved, as suggested by the Fermi-Pasta-Ulam experiments in the 1950s, that it has exact periodic and soliton solutions. The Toda lattice, as it came to be known, was then extensively studied as one of the completely integrable (differential-difference) non-linear equations which admit exact solutions in terms of theta functions of hyperelliptic curves. In this paper, we extend Toda's original approach to give hyperelliptic solutions of the Toda lattice in terms of hyperelliptic Kleinian (sigma) functions for arbitrary genus. The key identities are given by generalized addition formulae for the hyperelliptic sigma functions (J.C. Eilbeck \textit{et al.}, {\it J. reine angew. Math.} {\bf 619}, 2008). We then show that periodic (in the discrete variable, a standard term in the Toda lattice theory) solutions of the Toda lattice correspond to the zeros of Kiepert-Brioschi's division polynomials, and note these are related to solutions of Poncelet's closure problem. One feature of our solution is that the hyperelliptic curve is related in a non-trivial way to the one previously used.