2021/05/11 by Zhijie Chen, Chen, Zhijie, Ting-Jung Kuo +3
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Algebraic structures and combinatorial models #Advanced Differential Equations and Dynamical Systems
paper · pdf · doi:10.48550/arxiv.2105.04734
In this paper, we prove Dahmen and Beukers' conjecture that the number of integral Lamé equations with index n modulo scalar equivalence with the monodromy group dihedral DN of order 2N is given by Ln(N)=(1)/(2)( (n(n+1)Ψ(N))/(24)-( an% ϕ(N)+bnϕ(\tfracN2) ) ) +\frac2% 3εn(N). Our main tool is the new pre-modular form Zr,s(n)(τ) of weight n(n+1)/2 introduced by Lin and Wang \citeLW2 and the associated modular form Mn,N(τ):=∏(r,s)Zr,s(n)(τ) of weight Ψ(N)n(n+1)/2, where the product runs over all N-torsion points (r,s) of exact order N. We show that this conjecture is equivalent to the precise formula of the vanishing order of Mn,N(τ) at infinity: v∞(Mn,N(τ))=anϕ(N)+bnϕ( N/2). This formula is extremely hard to prove because the explicit expression of Zr,s(n)(τ) is not known for general n. Here we succeed to prove it by using certain Painlevé VI equations. Our result also indicates that this conjecture is intimately connected with the problem of counting pole numbers of algebraic solutions of certain Painlevé VI equations. The main results of this paper has been announced in \citeLin-CDM.