Number Theory Seminar

Shahidi's definition of the local coefficient of a representation relies on the existence of a unique model called the Whittaker model. The local coefficient allowed him to define the Langlands L-function of an automorphic representation at the so called "bad" primes, and establish certain predicted analytic properties for these L-functions such as the functional equation and meromorphic continuation. However, there are examples of automorphic representations that do not posses a Whittaker model, and so Shahidi's method does not apply. Friedberg and Goldberg were able to use another model to define a local coefficient; however, their result required a certain minimality hypothesis. Recently, Goldberg has used this technique to prove a crude functional equation for certain automorphic representations of $SO(5)$. I will describe the method of Friedberg and Goldberg and recent improvements to their result.