Computing the number of certain Galois representations mod $p$

Centeleghe, Tommaso Giorgio

Journal de théorie des nombres de Bordeaux, Tome 23 (2011) no. 3, pp. 603-627.

En utilisant le lien entre représentations galoisiennes et formes modulaires provenant de la Conjecture de Serre, nous calculons, pour tout premier $p\le 2593$, une borne pour le nombre de classes d’isomorphismes des représentations galoisiennes de $\mathbf{Q}$ sur un ${\overline{\mathbf{F}}}_p$–espace vectoriel de dimension deux qui sont irréductibles, impaires, et non–ramifiées en dehors de $p$.

Using the link between Galois representations and modular forms established by Serre’s Conjecture, we compute, for every prime $p\le 2593$, a lower bound for the number of isomorphism classes of Galois representation of $\mathbf{Q}$ on a two–dimensional vector space over ${\overline{\mathbf{F}}}_p$ which are irreducible, odd, and unramified outside $p$.

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