## Two-dimensional $$\ell$$-adic representations of the Galois group of a global field of characteristic $$p$$ and automorphic forms on $$\mathrm{GL}(2)$$.(Russian. English summary)Zbl 0585.12006

In this remarkable paper the author completes the proof of the Langlands conjecture concerning a one-to-one correspondence between irreducible two-dimensional $$\ell$$-adic representations of the Weil group of a global field $$k$$ of positive characteristic and irreducible infinite-dimensional automorphic cuspidal representations of $$\mathrm{GL}(2)$$ over the adele ring of $$k$$.
See also the previous papers of the author [Am. J. Math. 105, 85–114 (1983; Zbl 0536.14014); Proc. int. Congr. Math., Helsinki 1978, Vol. 2, 565–574 (1980; Zbl 0444.12004); Funkts. Anal. Prilozh. 15, No. 4, 75–76 (1981; Zbl 0479.14017)]. For a general discussion of Langlands’ program the reader may consult S. Gelbart [Bull. Am. Math. Soc., New Ser. 10, 177–219 (1984; Zbl 0539.12008)].

### MSC:

 11R39 Langlands-Weil conjectures, nonabelian class field theory 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings 11F70 Representation-theoretic methods; automorphic representations over local and global fields 14G15 Finite ground fields in algebraic geometry 11R58 Arithmetic theory of algebraic function fields

### Citations:

Zbl 0536.14014; Zbl 0444.12004; Zbl 0479.14017; Zbl 0539.12008