@article{BBHMWZ09,
author = {A. Bostan and S. Boukraa and S. Hassani and J. -M. Maillard
and J. -A. Weil and N. Zenine},
title = {Globally nilpotent differential operators and the square
{I}sing model},
journal = {J. Phys. A: Math. Theor.},
number = {12},
year = {2009},
volume = {42},
pages = {50pp},
doi = {10.1088/1751-8113/42/12/125206},
abstract = {We recall various multiple integrals related to the
isotropic square Ising model, and corresponding, respectively, to the
n-particle contributions of the magnetic susceptibility, to the (lattice)
form factors, to the two-point correlation functions and to their
lambda-extensions. These integrals are holonomic and even G-functions: they
satisfy Fuchsian linear differential equations with polynomial coefficients
and have some arithmetic properties. We recall the explicit forms, found in
previous work, of these Fuchsian equations. These differential operators are
very selected Fuchsian linear differential operators, and their remarkable
properties have a deep geometrical origin: they are all globally nilpotent,
or, sometimes, even have zero p-curvature. Focusing on the factorised parts
of all these operators, we find out that the global nilpotence of the factors
corresponds to a set of selected structures of algebraic geometry: elliptic
curves, modular curves, and even a remarkable weight-1 modular form emerging
in the three-particle contribution $\chi^{(3)}$ of the magnetic
susceptibility of the square Ising model. In the case where we do not have
G-functions, but Hamburger functions (one irregular singularity at 0 or
$\infty$) that correspond to the confluence of singularities in the scaling
limit, the p-curvature is also found to verify new structures associated with
simple deformations of the nilpotent property.},
}