Purely imaginary period matrices and rhombic period lattice for (3,3N+1)-curves

Determine whether, for the trigonal (3,3N+1)-curve V defined by the spectral equation −w^3 + w I_{2N−1}(z) + I_{3N+1}(z) = 0 (with h_2 = 0) whose finite branch points are all real or occur in complex-conjugate pairs, the first-kind and second-kind period matrices ω and η of the abelian differentials on V can be chosen purely imaginary and κ = η ω^{−1} real, and whether the period lattice exhibits the rhombic sublattice property Im ω'_j ∈ span{(1/2) Im ω_k : k = 1,…,g} for every j = 1,…,g.

Background

The paper constructs the Boussinesq hierarchy via the orbit method on coadjoint orbits in the loop algebra of sl(3) and identifies spectral curves as trigonal (3,3N+1)-curves of genus g=3N. Finite-gap solutions are expressed in terms of Kleinian σ/℘-functions associated with these curves, with periods determined by first- and second-kind differentials.

To obtain real-valued and possibly bounded solutions, the authors analyze reality conditions on the Jacobian. They conjecture that, when all finite branch points are real or come in complex-conjugate pairs, the period matrices can be put into a purely imaginary form and κ becomes real, implying strong symmetry of the period lattice. This structure underpins the expected real-analytic behavior of the Kleinian functions along specific real/imaginary subspaces of the Jacobian.