TY - JOUR
T1 - Functoriality for the classical groups over function fields
AU - Lomelí, Luis Alberto
N1 - Funding Information:
I cannot thank professor F. Shahidi enough for his support and encouragement. The majority of the results presented in this paper were obtained as a thesis written under his supervision at Purdue University. I would like to thank E. Goins, D. Goldberg, and J.-K. Yu for their comments upon reading an earlier version of the manuscript. In particular, J.-K. Yu has helped me clear up a couple of misconceptions on my part. I would also like to acknowledge the many useful conversations I have had with D. Goldberg, M. Krishnamurthy, and P. Kutzko. Finally, thanks are due to J. W. Cogdell for his interest in this work and to G. Henniart for mathematical communications. This project was finalized while partially supported by the University of Iowa, Department of Mathematics NSF VIGRE grant DMS-0602242.
PY - 2009
Y1 - 2009
N2 - Langlands' functoriality for generic representations from the split classical groups to an appropriate GLN is established. The functorial lift or transfer to GLN is obtained with the help of a converse theorem once the analytic properties of L-functions are studied using the Langlands-Shahidi approach. This paper is mostly devoted to understanding L-functions for the classical groups over a global function field, since the Langlands-Shahidi method has only been developed over number fields. To overcome many difficulties, stability of γ -factors under twists by highly ramified characters is used together with multiplicativity. Finally, by analyzing the image of functoriality, a proof of the Ramanujan conjecture for generic representations is obtained.
AB - Langlands' functoriality for generic representations from the split classical groups to an appropriate GLN is established. The functorial lift or transfer to GLN is obtained with the help of a converse theorem once the analytic properties of L-functions are studied using the Langlands-Shahidi approach. This paper is mostly devoted to understanding L-functions for the classical groups over a global function field, since the Langlands-Shahidi method has only been developed over number fields. To overcome many difficulties, stability of γ -factors under twists by highly ramified characters is used together with multiplicativity. Finally, by analyzing the image of functoriality, a proof of the Ramanujan conjecture for generic representations is obtained.
UR - http://www.scopus.com/inward/record.url?scp=79251547197&partnerID=8YFLogxK
U2 - 10.1093/imrn/rnp089
DO - 10.1093/imrn/rnp089
M3 - Article
AN - SCOPUS:79251547197
VL - 2009
SP - 4271
EP - 4335
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
SN - 1073-7928
IS - 22
ER -