STNB2022 (35th edition)
Valution of coefficients of polynomials geometrically realizing $\mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z})$
Presenters
Zoé Yvon
Abstract
The aim of the inverse Galois problem is to find fields extensions which Galois group is isomorphic to a given group. Here, we are interested in groups $\mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z})$ where $n$ is either a prime or a product of two distinct primes. We know that we can realize these groups as the Galois group of a given number field $K$, using the torsion points on an elliptic curve. Specifically, a theorem of Reverter and Vila gives, for each prime $n$, a family of polynomials, depending on an elliptic curve, whose Galois group is $\mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z})$. We determine a lower bound on the valuations of the coefficients of these polynomials, depending only on $n$, when the coefficients of the elliptic curve are taken integral.
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