We investigate the behaviour of elliptic Feynman integrals under modular transformations. This has a practical motivation: Through a suitable modular transformation we can achieve that the nome squared is a small quantity, leading to fast numerical evaluations. Contrary to the case of multiple polylogarithms, where it is sufficient to consider just variable transformations for the numerical evaluations of multiple polylogarithms, it is more natural in the elliptic case to consider a combination of a variable transformation (i.e. a modular transformation) together with a redefinition of the master integrals. Thus we combine a coordinate transformation on the base manifold with a basis transformation in the fibre. Only in the combination of the two transformations we stay within the same class of functions.

我们研究了椭圆Feynman积分在模块化变换下的行为。这有一个实践动机：通过适当的模块化转换，我们可以实现Nome平方的数量少，从而可以快速进行数值评估。与多个多对数的情况相反，对于多对数的数值评估，仅考虑变量变换就足够了，在椭圆形情况下，考虑将变量变换（即模数变换）与重新定义主积分。因此，我们将基础流形上的坐标变换与纤维中的基础变换结合在一起。仅在两个转换的组合中，我们才处于同一类函数中。