区分模p的椭圆曲线并识别乘积表示的像
Distinguishing elliptic curves modulo $p$ and identifying images of product representations
AI总结：
该研究针对定义在ℚ上的两条椭圆曲线与有理素数p，借助古尔萨引理刻画其剩余伽罗瓦表示乘积的所有可能像，定义见证比率并阐明其计算效用及与斯特姆界的关联。
AI中文摘要：
给定两条定义在ℚ上的椭圆曲线和一个有理素数p，我们研究它们的剩余伽罗瓦表示的乘积。利用古尔萨引理，我们明确枚举并完全刻画这类乘积表示的所有可能像。我们还为这些像群定义了相关不变量，称之为“见证比率”，并解释其计算效用以及与用于检验模形式同余的著名斯特姆界的关系。
英文摘要：
Given two elliptic curves defined over $\mathbb{Q}$ and a rational prime $p$, we study the product of their residual Galois representations. Using Goursat's lemma, we explicitly enumerate and completely characterize all possible images of such product representations. We also define associated invariants to these image groups, which we call \textit{witness ratios}, and we explain their computational utility and their relationship to the well-known Sturm bound for testing congruences between modular forms.
