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We study the Toda equations in the continuous level, discrete level and ultradiscrete level in terms of elliptic and hyperelliptic $\sigma$ and $\psi$ functions of genera one and two. The ultradiscrete Toda equation appears as a discrete-valuation of recursion relations of $\psi$ functions.doi:10.2991/jnmp.2003.10.4.8 fatcat:qujdll6jizfirll6cvmejuef3i