正方格伊辛模型的高阶Fuchsian方程：$\tildeχ^{(5)}$
High order Fuchsian equations for the square lattice Ising model: $\tildeχ^{(5)}$
摘要：
我们考虑的Fuchsian线性微分方程（模一个素数）$\tildeχ^{(5)}$，五粒子贡献的正方形晶格伊辛模型的磁化率。我们表明，人们可以理解的因式分解相应的线性微分算子的计算，只使用一个素数。从$\tildeχ^{(5)}$中去掉$\tildeχ^{(1)}$和$\tildeχ^{(3)}$的一个特殊线性组合，得到的级数被高阶全局幂零线性常微分方程零化.相应的（最低阶）线性微分算子，阶数为29，分裂成小阶的因子。一个五阶线性微分算子作为“耗尽”微分算子的最左边的因子出现，并且它被证明等价于$L_E$的对称四次幂，线性微分算子对应于椭圆积分$E$。这个结果推广了我们对低阶项$\tildeχ^{(3)}$和$\tildeχ^{(4)}$的发现。我们猜想，对于所有的$\tildeχ^{(n)}$，在最小阶线性微分算子中，存在一个等价于$L_E$的对称$(n-1)$次幂的线性微分算子作为最左因子.
英文摘要：
We consider the Fuchsian linear differential equation obtained (modulo a prime) for $\tildeχ^{(5)}$, the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination of $\tildeχ^{(1)}$ and $\tildeχ^{(3)}$ can be removed from $\tildeχ^{(5)}$ and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth order linear differential operator occurs as the left-most factor of the "depleted" differential operator and it is shown to be equivalent to the symmetric fourth power of $L_E$, the linear differential operator corresponding to the elliptic integral $E$. This result generalizes what we have found for the lower order terms $\tildeχ^{(3)}$ and $\tildeχ^{(4)}$. We conjecture that a linear differential operator equivalent to a symmetric $(n-1)$-th power of $L_E$ occurs as a left-most factor in the minimal order linear differential operators for all $\tildeχ^{(n)}$'s.
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