摘要翻译：
我们引入了一些重积分，这些积分的奇异性与$n$-粒子对正方形晶格Ising模型磁化率的贡献$\chi^{(n)}$的奇异性相同。在$n=1,2,3,4$的情况下，我们找到了由这些多重积分所满足的Fuchsian线性微分方程，并且在$n=5,6$的情况下，只对一些素数进行模，从而提供了$\chi^{(n)}$的一大组（可能的）新奇点。通过求解Landau条件，讨论了这些多重积分的奇异性结构。我们发现相关联的ODE的奇点与前导的Pinch-Landau奇点是一致的($n=6$)。得到的第二个显著特征是与多重积分相关联的ODE的奇异性归结为与{em有限个一维积分}相关联的ODE的奇异性。在所发现的奇点中，我们强调了这样一个事实，即在$\chi^{（3）}$线性微分方程中出现的二次多项式条件$1+3w+4w^2=0$实际上对应于所选椭圆曲线的一个显著性质，即复数乘法的出现。给出了椭圆曲线的复乘作为重整化群中所选生成元的复不动点的解释，即椭圆曲线的等均一性。在我们的多重积分中出现的大多数其他奇点与复杂的乘法情况无关，这表明了超出椭圆曲线理论的（motivic）数学结构的解释。
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英文标题：
《Singularities of $n$-fold integrals of the Ising class and the theory of
  elliptic curves》
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作者：
S. Boukraa, S. Hassani, J.-M. Maillard, N. Zenine
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最新提交年份：
2007
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分类信息：

一级分类：Physics        物理学
二级分类：Mathematical Physics        数学物理
分类描述：Articles in this category focus on areas of research that illustrate the application of mathematics to problems in physics, develop mathematical methods for such applications, or provide mathematically rigorous formulations of existing physical theories. Submissions to math-ph should be of interest to both physically oriented mathematicians and mathematically oriented physicists; submissions which are primarily of interest to theoretical physicists or to mathematicians should probably be directed to the respective physics/math categories
这一类别的文章集中在说明数学在物理问题中的应用的研究领域，为这类应用开发数学方法，或提供现有物理理论的数学严格公式。提交的数学-PH应该对物理方向的数学家和数学方向的物理学家都感兴趣；主要对理论物理学家或数学家感兴趣的投稿可能应该指向各自的物理/数学类别
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一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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一级分类：Physics        物理学
二级分类：High Energy Physics - Theory        高能物理-理论
分类描述：Formal aspects of quantum field theory. String theory, supersymmetry and supergravity.
量子场论的形式方面。弦理论，超对称性和超引力。
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一级分类：Mathematics        数学
二级分类：Mathematical Physics        数学物理
分类描述：math.MP is an alias for math-ph. Articles in this category focus on areas of research that illustrate the application of mathematics to problems in physics, develop mathematical methods for such applications, or provide mathematically rigorous formulations of existing physical theories. Submissions to math-ph should be of interest to both physically oriented mathematicians and mathematically oriented physicists; submissions which are primarily of interest to theoretical physicists or to mathematicians should probably be directed to the respective physics/math categories
math.mp是math-ph的别名。这一类别的文章集中在说明数学在物理问题中的应用的研究领域，为这类应用开发数学方法，或提供现有物理理论的数学严格公式。提交的数学-PH应该对物理方向的数学家和数学方向的物理学家都感兴趣；主要对理论物理学家或数学家感兴趣的投稿可能应该指向各自的物理/数学类别
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一级分类：Physics        物理学
二级分类：Computational Physics        计算物理学
分类描述：All aspects of computational science applied to physics.
应用于物理学的计算科学的各个方面。
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英文摘要：
  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.
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PDF链接：
https://arxiv.org/pdf/706.3367