Nataniel Marquis

In 1991, J.-M. Fontaine proved an equivalence between representations of the absolute Galois group $\mathcal{G}_{\mathbb{Q}_p}$ of the $p$-adic numbers with $\mathbb{Z}_p$ coefficients and certain semi-linear objects called $(\phi,\Gamma)$-modules. Stated like that, the previous sentence is both vague and crowded with of number theory concepts. Our first goal is to motivate the appearance of the first objects : first the Galois group, then the $p$-adic numbers and finally the representations themselves. Then, we would construct the semi-linear objects and try to grasp of the philosophy of this equivalence.