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Moduli problems, local conditions and the inverse Galois problem Arias-de-Reyna, Sara


The motivation for this talk comes from the inverse Galois problem for finite linear groups and its variations involving ramification conditions. Given an n-dimensional abelian variety $A/\Q$ which is principally polarised, we consider for each prime number the representation of the absolute Galois group of the rational numbers, $\rho_{A,\ell}: G_\Q \rightarrow \GSp(2n,\ell)$ attached to the $\ell$-torsion points of $A$. Provided the representation is surjective, we obtain a realisation of $\GSp(2n,\ell)$ as the Galois group of the finite extension $\Q(A[\ell])/\Q$, and the ramification type of a prime $p$ in this extension can be read off from the type of reduction of $A$ at $p$. This approach motivates our interest for the existence of abelian varieties over $\Q$ satisfying local conditions at a finite set of primes, which can be rephrased as the problem of the existence of a rational point in the intersection of a finite number of $p$-adic open sets of a suitable moduli space. In practice, it is very difficult to find such points, and a good strategy is to consider the Jacobian of curves in a family, which can easily be deformed p-adically. Recently there have appeared several constructions of curves defined over $\Q$ satisfying that the $\ell$-torsion representation attached to its Jacobian is surjective (e.g. Anni et al., Arias-de-Reyna et al. and Zywina for genus 3, more recently Anni and Dokchitser for any genus). In principle, these constructions allow one to carry out the above strategy in an explicit way to address the problem of producing realisations of $\GSp(2n,\ell)$ with prefixed ramification conditions.

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