Given an abelian surface, the number of its distinct decompositions into a product of elliptic curves has been described by Ma. Moreover, Ma himself classified the possible decompositions for abelian surfaces of Picard number $1 \leq \rho \leq 3$. We explicitly find all such decompositions in the case of abelian surfaces of Picard number $\rho= 4$. This is done by computing the transcendental lattice of products of isogenous elliptic curves with complex multiplication, generalizing a technique of Shioda and Mitani, and by studying the action of a certain class group on the factors of a given decomposition. We also provide an alternative and simpler proof of Ma's formula, and an application to singular K3 surfaces.

中文翻译：

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奇异阿贝尔曲面的分解

给定一个阿贝尔曲面，Ma 已经描述了其不同分解为椭圆曲线乘积的次数。此外，马本人对皮卡德数 $1 \leq \rho \leq 3$ 的阿贝尔曲面的可能分解进行了分类。我们在皮卡德数 $\rho= 4$ 的阿贝尔曲面的情况下明确地找到了所有这些分解。这是通过计算具有复数乘法的同质椭圆曲线的乘积的超越格，推广 Shioda 和 Mitani 的技术，并通过研究某个类群对给定分解的因子的作用来完成的。我们还提供了 Ma 公式的替代和更简单的证明，以及对奇异 K3 曲面的应用。