Explicit formulas for pre-modular forms Z_{r,s}^{(n,0,0)}(τ) in the Lamé case for n ≥ 5

Derive explicit expressions for the pre-modular forms Z_{r,s}^{(n,0,0)}(τ) associated with the classical Lamé equation y''(z) = [n(n + 1)℘(z; τ) + B] y(z) for integer n ≥ 5.

Background

The paper constructs pre-modular forms Z_{r,s}{(n)}(τ) that characterize monodromy data for generalized Lamé equations and provides explicit formulas for several low-degree cases, including the Lamé equation for small n (see (1.19)).

However, for larger Lamé parameters, deriving explicit closed-form expressions becomes increasingly difficult due to high weights and complexity. The authors note that beyond certain small values of n, explicit formulas have not been obtained.

References

For n ≥ 5, the explicit expression of Z r,s0,0)(τ )is not known so far.

— Monodromy of generalized Lame equations with Darboux-Treibich-Verdier potentials: A universal law  (2404.01879 - Chen et al., 2024) in Section 1 (Introduction), after Theorem 1.6 and the formulas in (1.19)

Several problems remain open. A first problem is to determine the transformation law of the accessory spectral polynomial $\Delta_N(q;a)$ under the full Lam\ e-preserving anharmonic subgroup and to identify invariant combinations of its roots.

— Some Results on Lame-Heun Spectral Geometry  (2609.24807 - Idiong, 21 Sep 2026) in Section 4, subsection “Open problems and further research”