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In this short note we give a new approach to proving modularity of $p$-adic Galois representations using a method of $p$-adic approximations. This recovers some of the well-known results of Wiles and Taylor in many, but not all, cases. A feature of the new approach is that it works directly with the $p$-adic Galois representation whose modularity is sought to be established. The three main ingredients are a Galois cohomology technique of Ramakrishna, a level raising result due to Ribet, Diamond, Taylor,...

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...