  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.
