超楕円σ関数による戸田格子の周期解と擬周期解について

Abstract

In this report, I summarize results in the paper (Kodama, Matsutani, Previato, Ann. Inst. Fourier 63 (2013) 655-688) to pose a problem to give an explicit relation between periodic and quasi-periodic solutions of Toda lattice. For a hyperelliptic curve Xg of genus g, we have a quasi-periodic solution of Toda lattice in terms of the hyperelliptic σ function and its addition theorem. Using the division polynomial of Xg, we find 2N-division points in its Jacobi variety and then have N-periodic solution of Toda-lattice. It is well-known that the N-periodic solution is associated with a hyperellptic curve ˆXg, N-1 of genus N-1 rather than g. However it is not clear how Xg and ˆXg, N-1 are connected geometrically, though the problem is very simple and natural. In this report, after I give a review of the recent development of σ function theory of higher genus and show the summary of our previous work, I give some comments on the problem.