In this paper we prove that for every integer d ≥ 1, there exists an explicit constant Bd such that the following holds. Let K be a number field of degree d, let q >max{d − 1, 5} be any rational prime that is totally inert in K and E any elliptic curve defined over K such that E has potentially multiplicative reduction at the prime 𝔮 above q. Then for every rational prime p > Bd, E has an irreducible mod p Galois representation. This result has Diophantine applications within the “modular method”. We present one such application in the form of an Asymptotic version of Fermat’s Last Theorem that has not been covered in the existing literature.

中文翻译：

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对数域进行乘法归约的椭圆曲线的 mod p Galois 表示的不可约性

在本文中，我们证明对于每个整数d ≥ 1, 存在一个显式常数乙d使得以下成立。让ķ是度数领域d， 让q >最大限度{d - 1, 5}是任何完全惰性的有理素数ķ和乙任何定义的椭圆曲线ķ这样乙在素数处有潜在的乘法归约𝔮更多q. 那么对于每个有理素数p > 乙d,乙有一个不可约的 modp伽罗瓦表示。该结果在“模块化方法”中具有丢番图应用。我们以现有文献中未涵盖的费马大定理的渐近版本的形式提出了一个这样的应用程序。