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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
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