  We introduce some multiple integrals that are expected to have the same
singularities as the singularities of the $ n$-particle contributions
$\chi^{(n)}$ to the susceptibility of the square lattice Ising model. We find
the Fuchsian linear differential equation satisfied by these multiple integrals
for $ n=1, 2, 3, 4$ and only modulo some primes for $ n=5$ and $ 6$, thus
providing a large set of (possible) new singularities of the $\chi^{(n)}$. We
discuss the singularity structure for these multiple integrals by solving the
Landau conditions. We find that the singularities of the associated ODEs
identify (up to $n= 6$) with the leading pinch Landau singularities. The second
remarkable obtained feature is that the singularities of the ODEs associated
with the multiple integrals reduce to the singularities of the ODEs associated
with a {\em finite number of one dimensional integrals}. Among the
singularities found, we underline the fact that the quadratic polynomial
condition $ 1+3 w +4 w^2 = 0$, that occurs in the linear differential equation
of $ \chi^{(3)}$, actually corresponds to a remarkable property of selected
elliptic curves, namely the occurrence of complex multiplication. The
interpretation of complex multiplication for elliptic curves as complex fixed
points of the selected generators of the renormalization group, namely
isogenies of elliptic curves, is sketched. Most of the other singularities
occurring in our multiple integrals are not related to complex multiplication
situations, suggesting an interpretation in terms of (motivic) mathematical
structures beyond the theory of elliptic curves.
