Synopsis
R. Langlands shows, in analogy with Artin's original treatment of one-dimensional p, that at least for tetrahedral p, L(s, p) is equal to the L-function L(s, π) attached to a cuspdidal automorphic representation of the group GL(2, /A), /A being the adéle ring of the field, and L(s, π), whose definition is ultimately due to Hecke, is known to be entire. The main result, from which the existence of π follows, is that it is always possible to transfer automorphic representations of GL(2) over one number field to representations over a cyclic extension of the field. The tools he employs here are the trace formula and harmonic analysis on the group GL(2) over a local field.
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