STNB 2016 (30è any)

Inverse Galois problem and uniform realizations


Samuele Anni


Let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$, let $n$ be a positive integer and let $\ell$ a prime number. Given a curve $C$ over $\mathbb{Q}$ of genus $g$, it is possible to define a Galois representation $\rho: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GSp}_{2g}(\mathbb{F}_\ell)$, where $\mathbb{F}_\ell$ is the finite field of $\ell$ elements and $\mathrm{GSp}_{2g}$ is the general symplectic group in $\mathrm{GL}_{2g}$, corresponding to the
action of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the $\ell$-torsion points of its Jacobian variety $J(C)$. If $\rho$ is surjective, then we realize $\mathrm{GSp}_{2g}(\mathbb{F}_\ell)$ as a Galois group over $\mathbb{Q}$.
In this talk I will descibe a joint work with Pedro Lemos and Samir Siksek, concerning the realization of $\mathrm{GSp}_6(\mathbb{F}_\ell)$ as a Galois group for infinitely many odd primes $\ell$. Moreover I will describe uniform realizations of linear groups.


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