Real subspaces of the Jacobian yielding real-valued Kleinian functions

Establish the existence of 2^g affine subspaces of the Jacobian of the (3,3N+1)-curve V, given by J^{Re} = {Ω + s | s ∈ R^g} with Ω equal to half-periods u[ε] for half-integer characteristics [ε], on which the Kleinian functions ℘_{i,j}(s+Ω) and ℘_{i,j,k}(s+Ω) are real-valued and possess poles; verify that, with periods chosen as in the preceding conjecture, these 2^g half-periods are purely imaginary (Ω ∈ i R^g), and that on the imaginary subspace J^{Im} = { i s | s ∈ R^g } the values satisfy ℘_{i,j}(i s) ∈ R and ℘_{i,j,k}(i s) ∈ i R.

Background

Building on the proposed purely imaginary structure of period matrices, the authors formulate precise loci in the Jacobian where the basic Kleinian functions determining the finite-gap solution attain real or purely imaginary values. These loci are affine subspaces translated by half-periods (related to characteristics) and the purely imaginary axis subspace.

Verifying these properties would provide concrete reality conditions for the finite-gap solutions w(x,t) and v(x,t) expressed via ℘-functions, enabling the construction of real-valued solutions on appropriate slices of the Jacobian and clarifying where poles may occur.