Monotonicity of t^{w−1}X_{w,1}(it) for extremal depth-1 quasimodular forms

Prove that for every even weight w ≥ 6 with w ≠ 8,10, the function t ↦ t^{w−1} X_{w,1}(it) is monotone decreasing for t > 0, where X_{w,1} denotes the normalized extremal quasimodular form of weight w and depth 1 for SL_2(Z).

Background

The paper develops a general framework to study monotonicity of functions of the form t ↦ tm F(it) for quasimodular forms F and derives several concrete monotonicity results for specific extremal quasimodular forms (e.g., X_{6,1}, X_{12,1}, X_{14,1}). Motivated by these results and numerical evidence, the author proposes a general monotonicity conjecture for all even weights w ≥ 6 except w = 8,10.

Establishing this monotonicity would extend the methods used to verify inequalities in proofs of optimality and universal optimality of certain lattices, providing a unified conceptual approach beyond the specific cases handled in the paper.

References

Based on Corollary \ref{cor:polymod_X61}, \ref{cor:polymod_X121}, \ref{cor:polymod_X141} and numerical experiments, we conjecture the following monotonicity property of extremal quasimodular forms of depth $1$. Conjecture For all even $w \ge 6$ but $w = 8, 10$, the function

t \mapsto t{w-1} X_{w, 1}(it)

is monotone decreasing for $t > 0$.

— Inequalities involving polynomials and quasimodular forms  (2602.10536 - Lee, 11 Feb 2026) in Conjecture, Section “Monotonicity of t^{w−1} X_{w,1}(it)”