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hal-00005796, version 1

## The Lamé family of connections on the projective line

Frank Loray () 1, Marius Van Der Put () 2, Félix Ulmer () 1

Abstract: A Lamé connection is a rank two connection $({\mathcal M},\nabla )$ on the projective line with simple poles at $0,1,t,\infty$ and local exponents $1/4,-1/4$ at each singular point. The relation with the elliptic curve $E$, given by the affine equation $w^2=z(z-1)(z-t)$, is investigated both in an algebraic and a complex analytic context. For the latter an analytic moduli space for Lamé equations is produced, based on monodromy above ${\mathbb P}^1\setminus \{0,1,t,\infty \}$ and the fundamental group $\pi _1(E)$. N.J.~Hitchin [Hi2] showed that the solutions of an irreducible Lamé connection can be written in terms of the Weierstrass zeta function. This was also noted by F.~Beukers (see [Be]) who presented a different proof. In the algebraic context a universal family, parametrized by a connected, non singular variety $P\rightarrow \{t\in {\mathbb C}\, |\, t\neq 0,1,\infty \}$ of relative dimension 2, of Lamé connections with free vector bundle ${\mathcal M}$ is explicitly computed. The Riemann-Hilbert correspondence yields an analytic isomorphism between the analytic moduli space and the algebraic one. In particular one observes that $P$ is not the full algebraic moduli space. The missing part is the {\it special family} of Lamé connections $({\mathcal M},\nabla )$ where $\mathcal M$ is not free but isomorphic to $O(1)\oplus O(-1)$. This family is computed. For every member of the family the differential Galois group of the Lamé connection is the infinite dihedral group $D^{{\rm SL}_2}_\infty$. Here $D_\infty^{{\rm SL}_2}$ and $D_n^{{\rm SL}_2}$ denote the preimages of the infinite dihedral group $D_\infty \subset {\rm PSL}_2$ and the finite dihedral group $D_n\subset {\rm PSL}_2$ under the map ${\rm SL}_2\rightarrow {\rm PSL}_2$. The closed locus $P_{reducible}\subset P$, representing the reducible Lamé systems, is explicitly computed. It turns out to be a non singular divisor. The differential Galois group of a Lamé connection corresponding to $p\in P$ is reducible for $p\in P_{reducible}$ and is an irreducible subgroup of the infinite dihedral group $D_\infty^{{\rm SL}_2}$ for $p\not \in P_{reducible}$. Let $P_N$ denote the constructible subset of $P$ consisting of the points such that the corresponding Lamé connection has the dihedral group $D_N^{{\rm SL}_2}$ as differential Galois group. An algorithm for the computation of the locus $P_N$, based on division polynomials for the elliptic curve $E$, the Painlevé VI equation and a transformation of Okamoto, is given. The loci $P_N$ with $N=2,3,4$ are explicitly computed. We note that the locus of the points in $P$ with differential Galois group equal to $D_\infty ^{{\rm SL}_2}$ is not constructible. The reason is that this linear algebraic group does not satisfy the condition posed in [Sin], (see also [B-vdP]). A weak point of our presentation of Lamé connections is the separation into two families, the one parametrized by $P$ and the other the special family. In the last section it is shown that one can produce a universal family containing both cases if one considers connection $({\mathcal M},\nabla )$ with ${\mathcal M}\cong O(0)\oplus O(-1)$. The {\it classical Lamé equation} ${\mathcal L}_{n,B}(y)=y''+\frac{f'}{2f}y'-\frac{n(n+1)z+B}{f}y=0 \ \mbox{ with } f=4(z-e_1)(z-e_2)(z-e_3),$ distinct $e_1,e_2,e_3$ and $e_1+e_2+e_3=0$, has regular singularities at the points $e_1,e_2,e_3$ with local exponents $0,1/2$. Further $\infty$ is a regular singularity with local exponents $-n/2,\, (n+1)/2$. For $n\in {\mathbb Z}$ the difference of the local exponents at $\infty$ lies in $1/2+{\mathbb Z}$. It is therefore possible to transform ${\mathcal L}_n$ into a Lamé connection if $n\in {\mathbb Z}$. A suitable transformation yields an isomorphism of ${\mathcal L}_{0,B}$ (with $B\neq 0$) with the special family. This explains in particular the well known result that $D_N^{{\rm SL}_2}$ with finite $N$ does not occur as differential Galois group for ${\mathcal L}_{0,B}$. By an algebraic transformation ${\mathcal L}_{n,B}$ (for $n\geq 1$) is mapped to a 2-dimensional subspace $S_n$ of the universal space $P$ of Lamé connections. The intersection of $S_n\cap P_N$ is equal to the finite set of points in the family ${\mathcal L}_{n,B}$ having differential Galois group $D_N^{{\rm SL}_2}$. For $n=1$ and $N=2,3$ our computations agree with the results of [Chi] and [Be-Wa]. In connection with this we observe the following. Any scalar equation for the Lamé connection, corresponding to a point $p\in P$, has, in general, an apparent singularity, i.e., a singular point different from $0,1,t,\infty$. This explains that the dimension of $P$ is larger than the dimension of the moduli spaces for the ${\mathcal L}_{n,B}$. Moreover, it is rather exceptional that a point $p\in P_N$ induces a classical Lamé equation with differential Galois group $D_N^{{\rm SL}_2}$.

• 1:  Institut de Recherche Mathématique de Rennes (IRMAR)
• CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
• 2:  University of Groningen
• University of Groningen
• Domain : Mathematics/Differential Geometry
Mathematics/Algebraic Geometry
• Keywords : géométrie algébrique – géométrie différentielle
• Internal note : 2005-23
• Available versions :  v1 (2005-07-04) v2 (2006-03-28)

• hal-00005796, version 1
• oai:hal.archives-ouvertes.fr:hal-00005796
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• Submitted on: Monday, 4 July 2005 14:24:09
• Updated on: Monday, 4 July 2005 14:30:01