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M. Toda in 1967 (\textit{J. Phys. Soc. Japan}, \textbf{22} and \textbf{23})considered a lattice model with exponential interaction and proved, assuggested by the Fermi-Pasta-Ulam experiments in the 1950s, that it has exactperiodic and soliton solutions. The Toda lattice, as it came to be known, wasthen extensively studied as one of the completely integrable(differential-difference) non-linear equations which admit exact solutions interms of theta functions of hyperelliptic curves. In this paper, we extendToda's original approach to give hyperelliptic solutions of the Toda lattice interms of hyperelliptic Kleinian (sigma) functions for arbitrary genus. The keyidentities are given by generalized addition formulae for the hyperellipticsigma functions (J.C. Eilbeck \textit{et al.}, {\it J. reine angew. Math.} {\bf619}, 2008). We then show that periodic (in the discrete variable, a standardterm in the Toda lattice theory) solutions of the Toda lattice correspond tothe zeros of Kiepert-Brioschi's division polynomials, and note these arerelated to solutions of Poncelet's closure problem. One feature of our solutionis that the hyperelliptic curve is related in a non-trivial way to the onepreviously used.

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