Abstract:
A Lamé connection is a logarithmic $\mathrm{sl}(2,\mathbb C)$-connection
$(E,\nabla)$ over an elliptic curve $X\colon \{y^2=x(x-1)(x-t)\}$,
$t\neq 0,1$, having a single pole at infinity. When this connection is
irreducible, we show that it is invariant under the standard involution
and can be pushed down to a logarithmic $\mathrm{sl}(2,\mathbb C)$-connection
on $\mathbb P^1$ with poles at $0$, $1$, $t$ and $\infty$. Therefore the
isomonodromic deformation $(E_t,\nabla_t)$ of an irreducible Lamé
connection, when the elliptic curve $X_t$ varies
in the Legendre family, is parametrized by a solution $q(t)$ of the
Painlevé VI differential equation $\mathrm{P}_{\mathrm{VI}}$.
The variation of the underlying vector bundle $E_t$ along the deformation
is computed in terms of the Tu moduli map: it is given by another solution
$\tilde q(t)$ of $\mathrm{P}_{\mathrm{VI}}$, which is related to $q(t)$
by the Okamoto symmetry $s_2 s_1 s_2$ (Noumi–Yamada notation). Motivated
by the Riemann–Hilbert problem for the classical Lamé equation, we
raise the question whether the Painlevé transcendents do have poles.
Some of these results were announced in [6].
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