Let F be a totally real number field which is unramified at $3,5$, and 7. We give explicit conditions on a finite place $v_0$ of F such that any elliptic curve over F with potential good reduction at $v_0$ is modular. The main point is to prove that there exist no elliptic curves with reducible mod 5, and 7 representations (or mod 3, 5, and 7 representations) under our assumptions.

中文翻译：

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椭圆曲线的可约模 105 表示和模块化

令F是一个完全实数域，它在$3,5$，以及7.我们在有限的地方给出明确的条件 $v_0$的˚F使得在任何椭圆曲线˚F与潜在的还原度好$v_0$是模块化的。重点是要证明在我们的假设下不存在具有可约模 5 和 7 表示（或模 3、5 和 7 表示）的椭圆曲线。