Frobenius modules are difference modules with respect to a Frobenius operator. Here we show that over non-archimedean complete differential fields Frobenius modules define differential modules with the same Picard-Vessiot ring and the same Galois group schemes up to extension by constants. Moreover, these Frobenius modules are classified by unramified Galois representations over the base field. This leads among others to the solution of the inverse differential Galois problem for $p$-adic differential...

This paper contains an application of Langlands’ functoriality principle to the following classical problem: which finite groups, in particular which simple groups appear as Galois groups over $\mathbb{Q}$? Let $\ell$ be a prime and $t$ a positive integer. We show that that the finite simple groups of Lie type $B_n\left({\ell }^k\right)=3DS{O}_{2n+1}{\left({𝔽}_{{\ell }^k}\right)}^{der}$ if $\ell \equiv 3,5(mod8)$ and $G_2\left({\ell }^k\right)$ appear as Galois groups over $\mathbb{Q}$, for some $k$ divisible by $t$. In particular, for each of the two Lie types and fixed $\ell$ we construct infinitely many Galois groups but we do not have a precise control...