论文标题
多项式意识到椭圆曲线的Galois表示图像
Polynomials realizing images of Galois representations of an elliptic curve
论文作者
论文摘要
逆加洛伊斯问题的目的是找到一个给定场的扩展，其galois组与给定组是同构的。在本文中，我们对n是整数的GL（2，z/nz）的子组感兴趣。我们知道，通常使用椭圆曲线上的扭转点，我们可以将这些组作为给定数字场的Galois组。具体而言，Reverter和Vila（2000）的定理给出了每个原始n的多项式，具体取决于椭圆曲线，其Galois组为GL（2，Z/nz）。在本文中，我们将该定理概括为多个方向，尤其是n non必然质量的。我们还确定了最低量的估值在我们的结构中产生的多项式系数，仅取决于n。
The aim of the inverse Galois problem is to find extensions of a given field whose Galois group is isomorphic to a given group. In this article, we are interested in subgroups of GL(2,Z/nZ) where n is an integer. We know that, in general, we can realize these groups as the Galois group of a given number field, using the torsion points on an elliptic curve. Specifically, a theorem of Reverter and Vila (2000) gives, for each prime n, a polynomial, depending on an elliptic curve, whose Galois group is GL(2,Z/nZ). In this article, we generalize this theorem in several directions, in particular for n non necessarily prime. We also determine a minimum for the valuations of the coefficients of the polynomials arising in our construction, depending only on n.
