High-order fuchsian equations for the square lattice Ising model: χ⁽⁶⁾

This paper deals with χ(6), the six-particle contribution to the magnetic susceptibility of the square lattice Ising model. We have generated, modulo a prime, series coefficients for χ(6). The length of the series is sufficient to produce the corresponding Fuchsian linear differential equation (modulo a prime). We obtain the Fuchsian linear differential equation that annihilates the depleted series θ(6) = χ(6) - 2/3χ(4) ?+ 2/45 χ(2). The factorization of the corresponding differential operator is performed using a method of factorization modulo a prime, introduced in a previous paper. The depleted differential operator is shown to have a structure similar to the corresponding operator for χ(5). It splits into factors of smaller orders, with the left-most factor of order 6 being equivalent to the symmetric fifth power of the linear differential operator corresponding to the elliptic integral E. The right-most factor has a direct sum structure, and using series calculated modulo several primes, all the factors in the direct sum have been reconstructed in exact arithmetics.