Valuation of coefficients of polynomials geometrically realizing GL_2(Z/nZ)

Zoé Yvon
I2M, Aix-Marseille Université

Date(s) : 28/04/2022   iCal
11 h 00 min - 12 h 00 min

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 $\mathrm{GL}_2(\Z/n\Z)$ where $n$ an integer. 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 $\mathrm{GL}_2(\Z/n\Z)$. 
We determine a lower bound on the valuations of the coefficients of these polynomials, depending only on $n$, when we take a short Weierstrass equation for the elliptic curve.

Campus de Luminy, Marseille


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