  We present a general method for analytically factorizing the n-fold form
factor integrals $f^{(n)}_{N,N}(t)$ for the correlation functions of the Ising
model on the diagonal in terms of the hypergeometric functions
$_2F_1([1/2,N+1/2];[N+1];t)$ which appear in the form factor
$f^{(1)}_{N,N}(t)$. New quadratic recursion and quartic identities are obtained
for the form factors for n=2,3. For n= 2,3,4 explicit results are given for the
form factors. These factorizations are proved for all N for n= 2,3. These
results yield the emergence of palindromic polynomials canonically associated
with elliptic curves. As a consequence, understanding the form factors amounts
to describing and understanding an infinite set of palindromic polynomials,
canonically associated with elliptic curves. From an analytical viewpoint the
relation of these palindromic polynomials with hypergeometric functions
associated with elliptic curves is made very explicitly, and from a
differential algebra viewpoint this corresponds to the emergence of direct sums
of differential operators homomorphic to symmetric powers of a second order
operator associated with elliptic curve.
