Functoriality for the classical groups over function fields

Luis Alberto Lomeli, Purdue University

Abstract

In this work, Langlands functoriality for generic representations of the split classical groups over global function fields is established. The converse theorem of Cogdell and Piatetski-Shapiro allows the production of a transfer after studying L-functions using the approach of the Langlands-Shahidi method. Complications in local-global arguments in the function field case, present an obstacle for giving a general treatment of γ-factors, which are needed to define local L-functions, as Shahidi has previously done in the case of number fields. Because of this, the study only includes those L-functions for the classical groups which are required by the converse theorem in the proof of functoriality. An important result is the stability of γ-factors under highly ramified characters, where care must be taken in order to include the case of characteristic 2. Stability coupled with multiplicativity is used in order to get around the difficulties in defining local factors presented by the function field case. The analytic properties of L-functions are obtained from those of Eisenstein series and careful adaptations of the arguments given in the number field case. It is important to notice that twists by highly ramified characters need to be incorporated in order to establish many results.

Degree

Ph.D.

Advisors

Shahidi, Purdue University.

Subject Area

Mathematics

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