Proof of a conjecture of Dahmen and Beukers on counting integral Lamé equations with finite monodromy

Abstract

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 $D_N$ of order $2N$ is given by \[L_{n}(N)=\frac{1}{2}\left( \frac{n(n+1)Ψ(N)}{24}-\left( a_{n}% ϕ(N)+b_{n}ϕ(\tfrac{N}{2}) \right) \right) +\frac{2}% {3}\varepsilon_{n}(N).\] Our main tool is the new pre-modular form $Z_{r,s}^{(n)}(\tau )$ of weight $n(n+1)/2$ introduced by Lin and Wang \cite{LW2} and the associated modular form $M_{n,N}(\tau ):={\prod }_{(r,s)}{Z}_{r,s}^{(n)}(\tau )$ of weight $\Psi (N)n(n+1)/2$, 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 $M_{n,N}(\tau )$ at infinity: \[v_{\infty}(M_{n,N}(τ))=a_{n}ϕ(N)+b_{n}ϕ( N/2).\] This formula is extremely hard to prove because the explicit expression of $Z_{r,s}^{(n)}(\tau )$ 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 \cite{Lin-CDM}.

53 pages