  We study the full susceptibility of the Ising model modulo powers of primes.
We find exact functional equations for the full susceptibility modulo these
primes. Revisiting some lesser-known results on discrete finite automata, we
show that these results can be seen as a consequence of the fact that, modulo
2^r, one cannot distinguish the full susceptibility from some simple diagonals
of rational functions which reduce to algebraic functions modulo 2^r, and,
consequently, satisfy exact functional equations modulo 2^r. We sketch a
possible physical interpretation of these functional equations modulo 2^r as
reductions of a master functional equation corresponding to infinite order
symmetries such as the isogenies of elliptic curves. One relevant example is
the Landen transformation which can be seen as an exact generator of the Ising
model renormalization group. We underline the importance of studying a new
class of functions corresponding to ratios of diagonals of rational functions:
they reduce to algebraic functions modulo powers of primes and they may have
solutions with natural boundaries.
