Abstract
The simplest case of the Langlands functoriality principle asserts the existence of the symmetric powers of a cuspidal representation of over the adèles of , where is a number field. In 1978, Gelbart and Jacquet proved the existence of . After this, progress was slow, eventually leading, through the work of Kim and Shahidi, to the existence of and . In this series of articles we revisit this problem using recent progress in the deformation theory of modular Galois representations. As a consequence, our methods apply only to classical modular forms on a totally real number field; the present article proves the existence, in this “classical” case, of and .
Citation
Laurent Clozel. Jack A. Thorne. "Level-raising and symmetric power functoriality, III." Duke Math. J. 166 (2) 325 - 402, 1 February 2017. https://doi.org/10.1215/00127094-3714971
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