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arXiv2025

Coincidences of Division Fields of an elliptic curve defined over a number field

For an elliptic curve defined over a number field, the absolute Galois group acts on the group of torsion points of the elliptic curve, giving rise to a Galois representation in $\mathrm{GL}_2(\hat{\mathbb{Z}})$. The obstructions to the surjectivity of this representation are either local (i.e. at a prime), or due to nonsurjectivity on the product of local Galois images. In this article, we study an extreme case: the coincidence i.e. the equality of $n$-division fields, generated by the $n$-torsion points, attached to different positive integers $n$. We give necessary conditions for coincidences, dealing separately with vertical coincidences, at a given prime, and horizontal coincidences, across multiple primes, in particular when the Galois group on the $n$-torsion contains the special linear group. We also give a non-trivial construction for coincidences not occurring over $\mathbb{Q}$.

math.NT↗

arXiv2023

Polynomials realizing images of Galois representations of an elliptic curve

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.

math.NT↗