论文标题
Mod P Galois表示椭圆形曲线的不可约性，其乘法降低了数字字段
Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields
论文作者
论文摘要
在本说明中，我们证明，对于每个整数$ d \ geq 1 $，都存在一个明确的常数$ b_d $，以便以下内容。令$ k $为$ d $的数字字段，让$ q> \ max \ {d-1,5 \} $是任何合理的素数，在$ k $中完全惰性，$ e $ $ e $任何定义在$ k $上的椭圆曲线，这样$ e $，以至于$ e $可能在prime $ \ mathfrak q $ qu $ q $ q $ qu $上可能具有乘数减少。然后，对于每个有理prime $ p> b_d $，$ e $都具有不可约的mod $ p $ galois代表。该结果在“模块化方法”中具有二磷剂的应用。我们以Fermat的最后一个定理的渐近版本的形式提出了一个这样的应用程序，该版本尚未在现有文献中涵盖。
In this note we prove that for every integer $d \geq 1$, there exists an explicit constant $B_d$ 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 $\mathfrak q$ above $q$. Then for every rational prime $p> B_d$, $E$ 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.
