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 .
Duke Math. J.
166(2):
325-402
(1 February 2017).
DOI: 10.1215/00127094-3714971
ACCESS THE FULL ARTICLE
It is not available for individual sale.