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arXiv · 2209.13477

Polynomials realizing images of Galois representations of an elliptic curve

Zoé Yvon
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Abstract

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.

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Zoé Yvon. 2022-09-27. Polynomials realizing images of Galois representations of an elliptic curve. https://doi.org/10.7169/facm%2F2106

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