Proof of a conjecture of Dahmen and Beukers on counting integral Lam\'{e} equations with finite monodromy

TL;DR

This paper proves Dahmen and Beukers' conjecture on counting integral Lamé equations with finite monodromy groups, using modular forms and Painlevé VI equations, revealing deep connections with algebraic solutions of Painlevé equations.

Contribution

It establishes the explicit formula for the number of such Lamé equations and links it to the vanishing order of a new modular form, employing Painlevé VI equations for the proof.

Findings

Confirmed the conjectured formula for counting Lamé equations.

Connected the problem to the vanishing order of a modular form.

Linked the counting problem to algebraic solutions of Painlevé VI.

Abstract

In this paper, we prove Dahmen and Beukers' conjecture that the number of integral Lam\'{e} equations with index $n$ modulo scalar equivalence with the monodromy group dihedral $D_{N}$ of order $2N$ is given by \[L_{n}(N)=\frac{1}{2}\left( \frac{n(n+1)\Psi(N)}{24}-\left( a_{n}% \phi(N)+b_{n}\phi(\tfrac{N}{2}) \right) \right) +\frac{2}% {3}\varepsilon_{n}(N).\] 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: \[v_{\infty}(M_{n,N}(\tau))=a_{n}\phi(N)+b_{n}\phi( N/2).\] 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 Painlev\'{e} VI equations. Our result also indicates that this conjecture is intimately connected with the problem of counting pole numbers of algebraic solutions of certain Painlev\'{e} VI equations. The main results of this paper has been announced in \cite{Lin-CDM}.