Let $E$ be an elliptic curve over a number field $K$. If for almost all primes of $K$, the reduction of $E$ modulo that prime has rational cyclic isogeny of fixed degree, we can ask if this forces $E$ to have a cyclic isogeny of that degree over $K$. Building upon the work of Sutherland, Anni, and Banwait-Cremona in the case of prime degree, we consider this question for cyclic isogenies of arbitrary degree.

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复合度同构的局部全局原则

令$ E $为数字字段$ K $的椭圆曲线。如果对于$ K $的几乎所有素数，素数具有固定度的有理循环等值线的$ E $模的减少，我们可以问这是否迫使$ E $超过$ K $具有该度数的循环等值线。在素数的情况下，以Sutherland，Anni和Banwait-Cremona的工作为基础，我们将这个问题视为任意度数的循环同构问题。