Two-dimensional $l$-adic representations of the Galois group of a global field of characteristic $p$ and automorphic forms on $GL(2)$

It is known that to each cuspidal automorphic representation of $GL(2)$ over the adele ring of a global field $k$ of characteristic $p$ there corresponds an irreducible two-dimensional $l$-adic representation of the Galois group of $k$. In the present paper it is proved that to each irreducible two-dimensional $l$-adic representation of the Galois group there corresponds a cuspical automorphic representation of $GL(2)$ over the adele ring. Thus the proof of the Langlands conjecture for $GL(2,k)$ is completed.