Existence of τ satisfying the simultaneous conditions 3η1(τ) + 2g2(τ) = 0 and 6e_k(τ) + g2(τ) = 0

Determine whether there exists τ in the upper half-plane such that both 3η1(τ) + 2g2(τ) = 0 and 6e_k(τ) + g2(τ) = 0 hold for some k ∈ {1, 2, 3}, which would lead to algebraic multiplicity d(−3e_k) = 5 for the n = (2,0,0,0) Lamé case.

Background

Using the universal law, the authors analyze the algebraic multiplicity of (anti)periodic eigenvalues for the Hill operator with DTV potentials. In the Lamé case n = (2,0,0,0), they classify multiplicities based on derivatives of e_k(τ) and certain relations among modular quantities.

They identify the simultaneous conditions 3η1(τ) + 2g2(τ) = 0 and 6e_k(τ) + g2(τ) = 0 (equation (7.7)) as yielding multiplicity 5, but the existence of τ meeting both conditions is unresolved.